If you do not want to watch the whole manual, it looks like this:
Now here's the explanation. The vertical axis of the left diagram depicts the cutoff and the bottom one the desired speed. Simple? A sort of that.
The second diagram is little bit complicated. The bottom line depicts the grade in thousands part of elevation per each unit of path, the vertical one - the desired speed (blue) and cutoff (pink). What does it mean? When we ride on grade, let's say 35/1000, then the speed obviously won't be that big. And in the same time, the cutoff position will be bigger by x, if running the same consist on grade x times bigger or at the same speed but x times lighter (because the grade actually affects on your speed/tractive effort ratio). So considering grade is proportional to your tractive effort (F = m*g*sin a).
Simple? Not at all.
May it be simpler? Yes.

Well, even having the diagram above it's not that clear what happens with the third part of equation - the throttle position. And here is the solution. Let's assume, that you have 2x more output power than needed. How may it affect your driving curves? Obviously it will mean that you can drive either 2x times faster on exact grade or haul 2x heavier consist.
As far as we have an extra 50% of power, we can convert it either in acceleration (what actually the measure of the tractive force) or in higher constant speed, so the curve may be much closer to the L shape.
This simple thought results in the next statement - if we run on lower cutoff positions, that let's say 2x smaller than needed, but the regulator is on 2x times bigger position, probably (actually it will be true till we reach the desired cutoff/speed position, when we will issue a steam pressure back force) we may get the same result.
What does it mean in practical applications (just in case we do have enough of extra power)? It means that on some cutoff position we can get the optimum mechanical power, that may be distributed either on force or speed, until we do not reach the max speed for this cutoff/throttle ratio and then issue the back force stopping us to maintain the higher speed. The same can be (but not always) said about single locomotive: running at full throttle and low cutoff may act as running at low throttle and big cutoff. Is it simpler? A little more, but the explanation is much more harder.

How it happens in real life (or in the TS)? Remember, in the first manual about the real life locomotives I said that the garden railroads are (often) going underpowered?
As you can see, according to the valve gear, the driver of the live steam locomotive gets the benefit of going underpowered so he can accelerate on certain cutoff value. This is real life explanation of all that I wrote above. So in order to maintain higher speed the driver in the video has to increase the throttle (and at speed when pack pressure appear lower the cutoff value), but if he wants to lower the speed he needs just to close the throttle.

As I noticed in the 1st part of the walkthrough, the best way to drive is to use the throttle and the cutoff both. Just because the switching them both is a little bit complicated, I suggest here more simple and compromise solution.
At first, you put the cutoff at 100% forward position and start to accelerate the train increasing the throttle. When reached 80% throttle, wait until lose approx 5 psi of max boiler pressure (to have a gap for non activating the security pressure valve) and then swap the position (100% throttle and 80% cutoff). Then slowly decrease the cutoff step-by-step according to next scheme.
Just because the power equals to max steam generatoion rate, you have not to go after it if you are not sure that you can recover the pressure.
In order to do that pull the cutoff to lower position until you got green indication. This means that you do not get a back pressure and your steam generation is equal to consumption. If you drive in cab, you will notice this effect simultaneously by changing of cylinders and steam chest pressure gauge (the steam pressure in cylinders will increase to certain value, specific to this steam locomotive, the chest pressure will decrease).
If the pressure indication became red repeat the process again. If you wouldn't reach the line speed if means you got not enough output power as a result of poor steam generation in the steam chest.
If it so, open the chimney doors and pull the blower in max position and slowly start doing the same but in opposite direction: switch the cutoff in red pressure zone and wait until it becomes green.
Watching the diagram you can easily notice that in most cases cutoff under 35, reached when doing the above algorithm is "undercut" so you have enough gap before you will reach back pressure effect.
So understanding what the game shows you may help you to understand what you should do.

Taking in account D.R.Y. suggestion from the second part of this manual, there are sometimes very significant differences between the simple driving, driving the passenger trains and driving the freight trains. In order to see it, let's compare styles of driving when start moving freight and passenger trains.
In passenger trains movement, just because of often stops and higher average speeds, the output power is usually utilized to achieve max line speed in relatively short time. Just because of low inertia and moving the consists with relatively short brake respond time (compare the time to achieve line speed and brake distance for 1 mile long few thousand tons freight train and the same values for light 0.3 mile long passenger train) like the passenger trains, you mostly convert the output power in big speed, low traction effort and middle traction effort while accelerating, in the same time converted in rise of speed. The "underpowering" of the steam train (short freight and fast passenger) can help to achieve the line speed, using low firemass, and two modes of the steam cutoff and throttle positions changing.
Just because this style of movement is often common in express freight traffic, having lengths comparable to lengths of normal passenger trains, we will cover them also, just for educational purposes.

The mode of driving reminds simplified mode, discussed in previous paragraph, though it has some slight changes, similar at the same time to cutoff changing and cutoff fixed style (discussed in next paragraph). Going up to 30 mph is always fixed and strongly connected to the weight of the consist
- hauling cold locomotives or few cars 22% cutoff (variable throttle)
- hauling express freight train (~10 cars or so) 33% cutoff value
- hauling double express freight train (~20 cars or so) 40% cutoff value

After reaching the desired speed, the train can't accelerate anymore (we reached the speed-traction effort ratio that is equal to our output power generation and now feel the back pressure). Further acceleration may be done in few ways: switching to coasting mode, switching to passenger mode or switching to acceleration mode. The first variant is clear - we close the throttle and cutoff to maintain speed we already reached.
The second and third variants aren't that clear. Just because we can't easily accelerate that fast.

In passenger mode (low cutoff setting, full throttle setting) the power is mostly converted to speed, so low (but optimal for this speed) cutoff may cause deceleration just because we generate not enough traction effort to compensate gravity force (this effect may be noticed in low power mixed traffic locomotives, like Pannier class), just because the traction force depends also on the length of stroke (shorter the stroke - shorter the time, when the full force of steam is fully applied to surface of the steam pistons), so this type of acceleration may be used in relatively short trains, when the loss of pressure isn't that critical (we can easily restore it just as we accelerated)

So in order to go in accelerate mode instead, we decrease the cutoff per some little quantity and in the same time increase the throttle until we get constant decrease of pressure like 0.1 pressure unit per 15-30 sec (depends on volume of boiler). When we notice the overall value of cutoff/throttle change will remind us somehow "swap" of the throttle and cutoff when accelerating the passenger train, but the main difference, that in passenger the primary cutoff setting was non optimal and was conversion of traction effort in acceleration (F = m*a) with intensive increase of output power (that can be applied in pass train for short time but not in the freight train). Also the "swap" won't be relative that big. Sometimes, if we do have enough steam pressure, we can even overrun the back pressure effect, when the pressure of fresh steam can overrun the resistance of the used steam and the gravity force. In this case the train will still accelerate, but overall efficiency will be lower.

In order to understand how it works imagine the combination of passenger mode (constant full throttle, variable cutoff), simplified mode (constant cutoff, variable throttle) and complicated light freight train acceleration mode (variable cutoff and throttle).

In order to start, first of all, we need enough fire mass (fire mass -> steam generation -> traction power -> traction force -> acceleration), enough pressure and enough water (water is heavily consumed if you run the freight train).

Now lets's start. There are few stages, so breaking the driving mode apart in few sections will help us to see the analogies with the driving modes described above.

Stage 1 retracting the buffers (variable throttle, fixed cutoff)
Actually this stage is the same as acceleration of the passenger train. We put cutoff in highest setting and slightly start increasing the throttle slowly. We can't increase it too fast to avoid wheelslip. Now our goal is not to reach 80% throttle, but to keep acceleration in fair boundaries (not that slow and not that fast, as in passenger trains). When we notice first signs of pressure loss, here starts the second stage.

Stage 2 giving more speed (fixed throttle, variable cutoff)
Now we keep that setting of throttle and start to decrease the cutoff slightly just to compensate the steam generation (keeping the green values of the pressure). At the same time we can recover the water lever, shoveling the coal at the same time. When reached some speed we will notice, that the steam generation is increasing, and the acceleration nearly stopped. Is that bad? No, because now we know that this cutoff setting don't cause the back force.

Stage 3 pre-final increase of speed (variable throttle, fixed cutoff)
Now we do almost the same as in simplified driving with fixed cutoff, increasing the throttle until we get slight loss of pressure. In gives us extra points of speed and acceleration. When the speed stopped to increase again, we have to choose the way go after: little race or keeping the speed.

Stage 4a. Little race (variable throttle, variable cutoff -> fixed throttle, fixed cutoff)
Just in case we didn't reach the desired speed, we need to play with throttle AND cutoff in any way (increase cutoff and left the throttle or vice versa) to get the constant acceleration (only the presence of it) and left the calculated positions fixed. Even if the pressure may fall it doesn't matter because we have already some base non-zero speed (so we can calculate that effect of pressure loss may not last too long). Just we reached desired speed, now we change to keeping the speed.

Stage 4b. Coasting (variable throttle, variable cutoff -> variable throttle, fixed cutoff)
Now we have to do the same as in 4a, but in backwards direction. We need to play with throttle to get any noticeable acceleration (ideally 0 to keep the speed). Now we need to keep cutoff setting, just switching throttle notch up and notch down to keep the speed.

It's probably most challenging part in driving steam trains. Just let's consider we are approaching the speed restriction or touching the top of the hill and have enough time to apply brakes (let's say we have 1 miles to the beginning of the speed restriction). Depends on the speed depicted we can slow down drastically and even lose the speed or touch the restriction few hundred feet before the sign.
The main problem that different trains and different locomotives even sharing common braking system have different braking curves.
Let's make a little pause and watch carefully to the steam locomotive automatic (or vacuum) train brake handle positions. It has usually from 3 to 4 positions - release (fast recharging of the brake pipe), running (release with refilling the leaks), lap (maintaining the constant pressure, set by driver), few noticeable non-lapped positions (in Westinghouse brake they are called "service" and "full service") and emergency. Actually the full service and emergency act near similar, just because both of them if not lapped may fully apply the brakes.
Let's now resume. We may put the brake lever at running position and, after that, at any service position at one second or even smaller, put back on running or lap (doesn't really matter) and calculate can we achieve any noticeable braking effect and the time what was spent to apply the brakes. Depends on whether the brakes are rigid (vacuum) or soft (air), the time may differ.
If the brake force is undesirably high, release the brakes and try more shorter time (usually in vacuum brakes). If the brake force is not enough (usually in air brakes), try to put brake at first service position at longer time and calculate the braking force and speed loss. If you're more satisfied with the result, make a series of discharge-lap-discharge-lap, each of switching back and forth made with shorter timings. If you notice that the brake force is slightly bigger than you want, try to make slight release and put in running or lap position (if it is possible) or full release (if the brakes are not responding to the short release) and repeat the process, getting more optimal pressure setting. If you succeed, try to remember this setting and repeat it again when riding on similar grade, slowing down or stopping at similar distance.
Applying the brakes is a challenge because sometimes you may near miss with the brakes setup. If the brake force isn't enough and you don't want to use the brakes, you may also try to apply the handbrake on tender, on the caboose (if it is available) and even make use of the locomotive (air) brake.
Putting in full stop when approaching the station is sometimes easier because, than slowing down, because the game already give you the brake distance (end of platform or freight track in the marshalling yard) so your braking time and distance may be easily calculated using the HUD menu. But there is a downside. If you're running in tight timetable and likely arrive too early, you can't easily stop your freight train as you can do in american diesel locomotives. So you should already start to slow down seeing yellow or any similar restrictive aspect. You need to calculate the braking force that way, so you'll get exactly 20 mph on 0.3 mile distance to the signal with dander aspect (if it will be noticeable earlier, you may decide another brake setting, depends on the aspect, or release the brake if it's now green or yellow).

If you want to read more about the brakes, you can also read this manual (note, that the brake reaction time may be much bigger in longer consists, so you need to take it in account too):
https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=1426074149

When you started going uphill, here starts another challenge. Consider you are driving two-headed PRR K4 with heavy train uphill. What will you do to keep the speed? The right answer - nothing, and it's the first part of the answer. Remember the diagram from the first paragraph? I am providing it again just to be clear:
When the steam train is moving and meet the uphill, the desired speed can't be reached because you already put all the horses to your steam machines. 1 metric horse (with Le article because of the metric system) is equal to certain kilonewtons of traction effort amount multiplied by certain m/s of speed of your steel horse. If you try to keep the speed you reached before, you very likely will lose the pressure, just because the steam generation is constant and you can't increase the steam generation in one turn. So the right solution is to change the cutoff to higher position and decrease the throttle (or leave all as is).
If you already know that the uphill is not that long, you may try to keep a slight decrement of pressure each 15-30 (more is better) seconds if that gives you a slight increase of speed. Also keep in mind that you also have to turn off all the consumers of the steam and to refill the boiler, because the steam generation depends on the amount of the fresh water, covering the smoke pipes as well as the amount of fire mass in the chimney.
If the uphill is too long and you nearly fail to keep constant pressure, try to use the stokers and blowers even if it is not recommended and you will be wasting the steam on the downhill (the correct action is to make the pressure in the boiler rising). The pressure fall is critical because the top speed depends of the steam pressure which is actually the measure of your output power. Remember the analogy with the diesel locomotives going uphill at full throttle.

In most cases the steam generation is enough for not wasting the coal (or oil), so you can close the doors in the chimney and turn the blower off (actually in real life the blowers are still working to keep the process of fuel burning in the chimney on all area of the burning surface).
The cutoff and throttle may be set at low positions to maintain constant speed, neither increment, nor decrement. If did so, the pressure will start to increase slightly, so keep in mind, that the method described above won't work.
If the train running on slight downhill, the cutoff may be set at biggest setting and zero throttle so the locomotive may keep the speed getting the benefits of compression brake. Sometimes you can get benefit of wasting steam in "incorrect", big forward cutoff setting, so you will stop because of the back pressure in steam machines will compensate the gravity force, but usually it is not very likely in most cases (you may lose the chest pressure and now the low pressure with big cutoff setting will act as big pressure with low cutoff setting).

Train simulator has a lot of hotkeys, some act nearly the same as in the electric and diesel locomotives, but some are steam locomotive specific ones.
The most important key in the TS is ... Ctrl+A (at least in the Riviera lines route). It activates the in-cab assistant that makes advices what to switch and turn (like in the Diesel railcar simulator video game). Usually I'm playing witout it because I enjoy watching and hearing the steam machines working and making nice screenshots.
If you do not have a possibility to activate it, try to memorize all the keys:
F5 - activate/deactivate the detailed information about the values of your steam engine
C - open/close cylinder drain coX that allow you to blow water from the cylinders
M/Shift-M - activate/deactivate the dampers
I/Shift-I - firemans injecter
K/Shift-K - exhaust feedwater
O/Shift-O - live injecter
L/Shift-L - live feedwater
R/Shift-R - stoking

This manual was based upon reading the forums and theoretical background. I say a lot of thanks to LeadCatcher and Wolf for their suggestions and advices. I also will note anybody, who may give feedback and show the mistakes or uncertainties. Any help appreciated. I also say many thanks to people that encouradged me to do this work, just because I saw they have problems driving the steam trains.