Let's discuss classic and classic-styled first person shooters. I don't like modern ones, to be honest, and the reasons are is that they don't usually have the interesting weapons, cool weapons, interesting level design or sheer personality of modern day FPSes. So, let me explain to you a few of my personal favorites and why you should get them.
Doom: You can never forget this one! It's pretty much aged perfectly, it's easy to find, and there's tons of mods out there to play. Skulltag is also great for multiplayer.
Rise of the Triad: Basically a spiritual sequel to Wolfenstien 3D, the graphics and level design quite as good, but it's a much faster game with a few cool features. It's got 5 characters each with their own stats, and it also has a powerup that turns you into a dog that shoots exploding barks.
Duke Nukem 3D: One of the best. Don't let Duke Nukem Forever hurt your judgement of this game, it's seriously one of the best FPSes ever made. The best levels in the genre, Duke is still a pretty awesome character, and some great weapons, including the shrink ray. Also has tons of mods, if you look.
Painkiller: This is a much newer game than the other games I mentioned, but it's still awesome. It also has a gun that fires shurikens and lightning.
Shadow Warrior: If Duke Nukem 3D was basically every action movie mixed into one game, this is basically every kung fu movie mixed into one game. Same developers as Duke, same engine, but there's new features like vehicles.
Blood: Warning, as you'd except, this games is REALLY bloody, even more than the other games. Same engine as Shadow Warrior and Doom, only all your weapons come with alternate firing modes, a first for the engine. Also, you can decapitate zombies and kick their heads like soccer balls.
Descent: Kind of like Doom mixed with Star Fox. You control a spaceship that can move around in 360 degrees, travelling through space stations blowing up robots. It's also OK to play if your parents are wussy and afraid of violence.
If you want to more about any of these, check through Google for videos and such. I'd be happy to let you know about any of those, though.