Play Trivial mode(Level 1) using quick play.
Go back to your own ship after each game. Bug world will be easier in general.
I would still engage to enjoy different levels and different missions sometimes for fun.
However, this is the fastest way I know to help you unlocking the Patriot Exosuit.

If you are skillful player which is comfortable with high stress,
you can host Helldive difficult level with the "kill and thin the number" mission.
You might need to search through the globe for that specific mission.
You will get around 900 EXP with 6-8 mins game when you have 3 more nice players with you.
You do not need to walk between mission as it will only have one mission, which is to kill 400-500 enemies and then leave. The extraction is quicker than usual which is like 10 seconds.

1. You will need to earn a lot of EXP to be at level 25.
Unlike Helldiver 1, Patriot Exosuit is available for unlock only after level 25.
High EXP reward per minute is important. And the EXP reward in this game is not well balanced.
For example, shutting down output will only give you 5 basic EXP and it could take you 0.5-3 minutes. But finishing main mission will give you 100 basic EXP usually and it takes around 2-10 mins usually, which definitely would not cost you 20x time for 20x EXP

2. You will get EXP whenever you finish a game with at least 1 completed mission.
It does not matter if you join in the last 10 seconds or first 10 seconds.
You will have the same amount of EXP with everyone in the game once you are there in the game.
With quickplay, you have higher chance to get into mid-game or end-game.

3. In Trival mode, You will get around 200 EXP in easy pace 5mins game without encountering heavy armored enemy.
You may have shorter game time down to 1-2 mins if you join the ending of the game by luck.
With the load time and difficult mission, it will take you around 10mins to complete a game, top.
Challenging mode(lv 4) or above will usually only give you same 200+ EXP or less with much difficult enemies within similar time frame.
For most of the Extreme(lv 6) or above game, you will get around 800+ EXP with full 40mins hard core fight. (You might have fun if you dont care much about EXP)
You could end up fail the main mission sometimes and get very little or zero EXP at the end.

4. You will have better game expectation from your teammate as they are usually new players
They would not aim for finishing all the sub-missions or shutting down all the outputs.
They will not complaint if you call for extraction ASAP
They might be bad host as they dont know how to start the game or pick the map location.
But it is very rare that you are in the map phase when you join the game. They are usually in the mission already.

5. If you have helldiver(player) in the same game with you, you will have +20 EXP for each helldiver.
The EXP will be the same for each player and it is not like share pool with splitting EXP.
And quickplay usually tend to match you with larger player pool.

6. I tried to host game. The game crashes/freezes a few times in a day.
I tried to join game via selecting the game on the screen. I often get full room or session not available message even it was half full before.
In quickplay, I usually have much better multi-play matching experience.

7. If you complete the mission within remaining 80%, then you will have 80 EXP bonus.
If you have 60% time left, then you have 60 EXP bonus.
So, calling for extraction as soon as possible would increase your EXP.
Each game is about 40 mins unless you extract early. Usually you will take 2-3 mins to complete the only mission in trivial mode and fixed 2 mins to wait for the extraction.
80+ EXP bonus is a lot when you are earning only 100 EXP from the mission in the Trivial mode.
You can go to extraction site site first when your teammate is working on the final phase of the mission. It will save your 20 seconds to end each game usually.

7. Bugs world will be easier as they do melee attack more often.
Automation world will have ranged attack bots usually.

8. Please note that it is definitely not the best way or only way to experience the full game.
Please read the title again before you write the hateful comment.
High level sample could only be found in higher level mission.
Joining quickplay might result in less badges to buy items in general.
You do not have control on having second deployment. And the second deployment have better badge reward.

9. Last but not least, I do hope that helldiver 2 developer would change that EXP reward mechanism in the future and give the right people with right reward.
I also hope that Patriot Exosuit could be available early on and let newbies have better protection and enjoy the game easier.