Ah, the classic genre of First-Person Shooters; how you have devolved over the years. I can't stand most modern shooters, as I find cover-based shooting, regenerating health, two-weapon systems and everything else that modern games have screwed up annoying. So, I decided to enter the hypothetical time machine and go back to 1996. I bought Quake, and immediately I found it appealing.
Quake is fast-paced FPS at its finest. It is all about killing hordes of baddies with multiple unique weapons in large, semi-maze-like levels that look like actual effort was put into designing them, something you don't often see these days. The weapons are all enjoyable to use, and are all different from eachother. Enemies come in many varieties, and each enemy type requires its own strategy to take it down. And, this game is also rather difficult, even on "easy", but that's why there is a quick-save button.
Quake is a prime example of what made 90s shooters fun. It doesn't hold your hand the whole way through like modern games. No, it tells you to get to the exit, and leaves you to figure out how to get there. So, if you are looking for a FPS that will challenge you and entertain you equally, get Quake. It's only $10 on Steam, and it will run on pretty much any Windows computer.