Just getting back to the gaming scene thanks to GOG.
The need for a controller is that I have limited use of my left hand for WASD. Industrial accident removed first two fingers. Luckily, I got to keep the thumb.
Looking at the Logitech F310 Gamepad as apparently it can be used for older (Xinput) games and also current titles (Directinput).
I have absolutely no experience with a game controller as I always used the mouse/ keyboard combo exclusively.
So, your advice will be very important before deciding and making a purchase.
Games I've played and enjoyed in the past:
System Shock2
Most of the Quake titles
Unreal Titles.
Thanks
Scartch...

Get ready for 500 different answers, all of them insisting the other 499 answers are useless pieces of shit. :P

Don't listen to tinyE, his statement is useless.

Most of the games you mentioned aren't really made for a gamepad as a friendly warning. But the F310 served me quite well. No rumble feature and the sticks are a little on the slippery side.

My current controller is the Steelseries 3GC, but I'm actually waiting for the Steam Controller to be cracked open.

To give you a serious answer to your question: Sorry to hear about your accident. I don't know much about the games, but couldn't you remap the controls? Like, arrow keys with the right hand and mouse with the left hand?

The Logitechs are fine, but you'd probably also be able to use a 360 controller with some remapping software.
Since you can then map the controller to keyboard presses and mouse movements, it should work for most games.
I rarely use a controller though, so it's entirely possible I'm wrong.

If you're missing your pointer and middle finger, a controller might not work well for you because of the buttons on the top controller. That said. the Steam controller might be useful to you in theory as you can easily remap buttons and have a shift button to shift controller so you can avoid those buttons on the top of the controller as long as a game works in Steam, but I've encountered a few with problems with some titles. I'm not sure if it works properly with System Shock 2, for example. I'll wish you the best of luck anyway, though.

As for GOG games working in Steam, you can add links to the GOG games from Steam by clicking in the bottom left corner and selecting add non-Steam game.

A gamepad may be the better solution in this case indeed, since most of the movement is done by the thumb this way. Good idea.
Another craftsman here, I feel for you and have seen a lot of this too.

I'd go for a gamepad with a lot of buttons on the right side and as always I'd recommend the Saitek p2600.
It was actually made with FPS in mind although I use it for other purposes.
It has 6 action buttons for your right thumb and four shoulder buttons (which makes two for you on the right side).
I always recommend this gamepad anyway, since it's awesome for so many kinds of games and my 4 ones work really well.
It's out of production for many years, but you can still get it used sometimes, has become a bit more expensive due to the fact, that people have realized, that there is no comparable gamepad on the market anymore after the ones from Saitek and Logitech (Wingman) went out of production and made way for inferior ones with only four front action buttons (bad for action RPGs and the Capcom fighters like Street Fighter, Darkstalkers, X-Men, Marvel vs. Capcom, etc...).

Playing FPS with a gamepad needs exercise though. I can't do it (had no real reason to try it often anyway), but since the kids that didn't grow up with keyboard and mouse controls seem to prefer it, it should be possible.

Arcade sticks may work well for FPS with autoaim like Doom 1+2, Heretic and Hexen, but for more modern FPS like Unreal you'd need an analog second stick for aiming - gamepads have those.

I don't know of this would help the OP but now I'm in it for the sake of my own curiosity, do they make a Wii type controller for PCs?

You got that backwards. Direct Input is for older titles. Xinput is for newer ones.

Really appreciate your checking in to help me with this.
Should have mentioned that I am now located in a rural area. My internet connection doesn't have the bandwidth to handle Steam. I would imagine that a Steam controller would not be an option.
A controller that the buttons can be mapped on might work.
Have to do some more investigating.

That makes two of us.