I think this is the first TR I ever played, and only recently finished.

A huge improvement over the previous two titles, TR3 feels like it has more freedom.
Immediately after finishing the introduction levelset of India, you're offered a choice of your next adventure. Yes, there is a "right" order in which to do these locations, but in the end, it's all a matter of preference.
I always do Nevada first to ensure that I don't lose valuable weapons later on, due to that section having one of those iconic "you got no weapons for half the level, good luck" levels that every Tomb Raider seems to have.

Next on, is the story. Lara is after a thing, the thing appears to be only one of a set, Lara is given information on locations of the other things in the set, and that concludes India. Three locations later, you're freezing your butt off in Antarctica and, in the vein of The Thing, facing horrors there. Good luck surviving without proper equipment. Trust me, you'll want to do that location order if you want to avoid hard battles.

New mechanics include a freezing meter for being underwater, ducking through small gaps, sprinting, doing a diving roll after a sprint and monkey swinging.
They add charm to the gameplay, and it feels like Lara is a true athlete now.
Don't forget to try holding the walk button while pulling yourself up on a ledge!

Lara's arsenal grows considerably, now including a Desert Eagle, MP-5 which functions the same as the TR2 Carbine, and a Rocket Launcher.
New vehicles have also been added, a Quad Bike, a Canoe, an underwater vehicle and a stationary Rocket Turret.

Lara's Home got a huge makeover, allowing for testing out vehicle controls, trying out the targeting system for the guns and using Lara's skills in both a training and timed training environment. Certain secrets within the house require at least basic knowledge of the new mechanics to get to them.
Still can't kill Winston, though. He acts as target practice.

Cutscenes are pretty same-y compared to TR2, with voice acting being slightly better now. It's a little more dramatic, especially when watching action cutscenes.

The Gamma graphical settings are fun to play with. I suggest playing on 3 for the true experience, as 10 looks like sunlight coming from everywhere and nowhere at the same time.
Playing on 1 feels like going through a horror game.

10/10, I'm biased towards this game.

BONUS!
The expansion for this game is missing from this release, but you can grab it elsewhere and use it with this version, with a cost. Since you will be overwriting the expansion's .exe, you will be unable to beat the final boss, as the final boss in the expansion is the one boss from the base game invulnerable to your weapons.

Someone made a "remastered" version of the soundtrack, available for ingame usage. I suggest you get that, since nothing
has changed apart from some extra tunes and addons like reverb, modern instruments and slightly better note mapping.

I think this is the first TR I ever played, and only recently finished. A huge improvement over the previous two titles, TR3 feels like it has more freedom. Immediately after finishing the introduction levelset of India, you're offered a choice of your next adventure. Yes, there is a "right" order in which to do these locations, but in the end, it's all a matter of preference. I always do Nevada first to ensure that I don't lose valuable weapons later on, due to that section having one of those iconic "you got no weapons for half the level, good luck" levels that every Tomb Raider seems to have. Next on, is the story. Lara is after a thing, the thing appears to be only one of a set, Lara is given information on locations of the other things in the set, and that concludes India. Three locations later, you're freezing your butt off in Antarctica and, in the vein of The Thing, facing horrors there. Good luck surviving without proper equipment. Trust me, you'll want to do that location order if you want to avoid hard battles. New mechanics include a freezing meter for being underwater, ducking through small gaps, sprinting, doing a diving roll after a sprint and monkey swinging. They add charm to the gameplay, and it feels like Lara is a true athlete now. Don't forget to try holding the walk button while pulling yourself up on a ledge! Lara's arsenal grows considerably, now including a Desert Eagle, MP-5 which functions the same as the TR2 Carbine, and a Rocket Launcher. New vehicles have also been added, a Quad Bike, a Canoe, an underwater vehicle and a stationary Rocket Turret. Lara's Home got a huge makeover, allowing for testing out vehicle controls, trying out the targeting system for the guns and using Lara's skills in both a training and timed training environment. Certain secrets within the house require at least basic knowledge of the new mechanics to get to them. Still can't kill Winston, though. He acts as target practice. Cutscenes are pretty same-y compared to TR2, with voice acting being slightly better now. It's a little more dramatic, especially when watching action cutscenes. The Gamma graphical settings are fun to play with. I suggest playing on 3 for the true experience, as 10 looks like sunlight coming from everywhere and nowhere at the same time. Playing on 1 feels like going through a horror game. 10/10, I'm biased towards this game. BONUS! The expansion for this game is missing from this release, but you can grab it elsewhere and use it with this version, with a cost. Since you will be overwriting the expansion's .exe, you will be unable to beat the final boss, as the final boss in the expansion is the one boss from the base game invulnerable to your weapons. Someone made a "remastered" version of the soundtrack, available for ingame usage. I suggest you get that, since nothing has changed apart from some extra tunes and addons like reverb, modern instruments and slightly better note mapping.