Anything mentioned above and any game from The Elder Scrolls series (specially Morrowind)

It's not a classic, but had lots of fun with Mirror's Edge. Check out minimum reqs before buying it though. Other than that, the already mentioned games are all great; certainly you shouldn't miss System Shock 1 & 2, Thief 1 & 2, Undying, AvP as well as Deus Ex. I'd throw in Jedi Outcast too, if you're fan of SW.

BioShock series
Dying Light

I would recommend a hidden gem: TRI: Of Friendship and Madness. it has not got the recognition it deserves, it is a really good and clever first person puzzler

https://www.gog.com/game/tri


If you are looking for a more straight up shooter games, then I would also suggest Tower of Guns. the name says it all, really. it is a coffee break game, each run can take about 40 min, but it shines in replayability (you will not reach the top or the hidden true ending in first go anyway).

https://www.gog.com/game/tower_of_guns

I just started playing Hidden & Dangerous 2 last night. I'm not far enough yet to say whether it's really good or not, but based on reviews and the first campaign, it's definitely a contender. In any case, there aren't too many tactical squad based shooters like this and H&D2 seems to have aged reasonably well. I thought the first game was pretty good too but I played it years ago and I'd have to see whether that opinion still hold up today.

Thief Gold.

If you like RPGs and you haven't played it, Morrowind is a must-play.

I also second System Shock.

If you're looking for some more relaxing "experiences", there's also first person walking sims and adventure.

Off the top of my head:
- Firewatch
- The Vanishing of Ethan Carter
- What Remains of Edith Finch
- Conarium
- Ether One
- Layers of Fear series
- Observer (or: >observer_)

and then you have the FP horror games like

- SOMA
- Amnesia series
- Outlast series
- Penumbra

Please check system requirements first, some of these games could be too much.

I am sorry not to have responded earlier but i got banned for some reason

So anyway Cyberpunk 2077 has pretty low requirements. On low resolution and low graphics setting any potato will run it

Thanks for the recommendations. There's a sale now, and Fallout 3 and New Vegas got added to my collection!

Those

I have an intel HD 620, which is probably similar to yours.

Here are games that have run well, and that I have enjoyed:

Call of Juarez 2 and 3- good shooters
Skyrim Legendary edition (runs well on 1600x900 and high textures, if you turn all the other effects way down)- very fun game!
Fallout 3
Fallout New Vegas
Oblivion
Portal 2

If you wanna try something different to shooters:

Myst series
Legend of Grimrock series

Also I can't believe no one mentioned the 'legendary' Daikatana!!! Blasphemy!!! xD :P

There are a lot of first-person genres: rpg, shooter, platformer, immersive sim, adventure, survival horror, puzzle, walking sim, blobber, beat 'em up etc.

But to acknowledge my own agenda:

Thunderscape,
Wizards & Warriors,
and Might & Magic 6-9 are all interesting

Ravelofts
and Drizzt game exist (well, they are dandy, I suppose)

Paper Sorcerer is newer, but is basically a Wiz4 for mortals.

Frayed Knights, while not on GOG, deserves more love.

Wizardry 8 is the best crpg ever made, accept no substitutes, ... because there are none (proceeds to cry in a corner for half an hour.)

Thanks for all messages and recommendations.

I just finished Far Cry 2. I liked it a lot. It is a flawed masterpiece.

As for the graphics powers, this might not mean much but I was able to play in on native resolution (FHD) with the highest graphical settings and no performance degradation at all.

Now I started Deus Ex and it is very interesting, although a bit quirky.