Well, it kind of depends on how far you want to go into FPS games. There's Halo, Halo 2, Team Fortress 2, the Half-Life series, Battlefield, and well, lots of games out there rated M for the PC are shooters. There are a few T rated shooter games as well. If anything, if you are a Sci-Fi/killing aliens kind of person, lean more towards Halo. If you are a Star Wars fan, go with Star Wars: Battlefront II. If you enjoy playing with other people on the internet with a variety of weapons, go with Team Fortress 2. Modern Warfare, Call of Duty 4 and Battlefield, and if you like adventure/shooting aliens/Sci-Fi games, the Half-Life series. Otherwise, here's games I'd choose from if you can run them.
Star Wars: Battlefront II
Half-Life 2: Deathmatch (Online/LAN multiplayer)
Team Fortress 2 (Online/LAN multiplayer)
Day of Defeat
Company of Heroes
Garry's Mod (requires the Half-Life 2 engine or a source game of the mod, such as Half-Life 2, Counter Strike, etc).
Call of Duty 4: Modern Warfare
Crysis (high end PC needed!)
Call of Duty 2 (for another WWII game)
The Orange Box (Includes Team Fortress 2, the Half-Life games, and Portal and Counter Strike I believe).
Me being me, I'll recommend the older games, such as Deus Ex: Conspiracy. Its a fun game, quite a story to tell behind it. If you're a fan of 'horror' as some people called it, Doom 3 is a choice, with some nice multiplayer. All in all, if you just want a game to mow down other players without much of a story, then to Valve with you. Smith6612 already mentioned the games, such as Battlefront 2, and Team Fortress 2, but STALKER also has some nice FPS aspects, such as quality of your gun, its condition, accuracy, rate of fire, and so on.
If u plan to buy a PC FPS, I'd recommend Call of Duty 2 or Call of Duty 4. I still play these most of the time.
These are some good games I have and play occasionally or once in a while:
Ghost Recon Advanced Warfare 1 and 2
Rainbow Six Vegas 1 and 2
Call of Duty 1
Gears of War (Not Really a FPS)
Quake 3 and 4
Unreal Tournament 2004 and 3
F.E.A.R and its Expansions. (The Multiplayer is Free and U can find it in the Wikipedia List)
Halo 1 and 2
Speaking of Free FPSes, I'd recommend Sauerbraten and Nexuiz, which are on the wiki list. One of the things that makes Sauer special is its realtime map editing, and through a certain type of multiplayer game, everyone can add their own touch to the map. It allows for very unique maps
It's FREE & there's a real cash economy in game world. Very Interesting.
Another interesting point is that this game will soon be equipt with Cryengine 2 game engine. For those who doesn't know, it's one of the best game engines out. Game Crysis uses this engine and alot more to come... To name one.. Blue Mars...
I can't wait for Blue Mars to come out. This is what I', waiting for and SC2. Also, another MMORPG FPS game not out yet but soon is Huxley!
I'm recruiting so if you gamers interested in join you're all welcome to join.
Dude you should totally try F.E.A.R. Heard of it? I beat the game. Scared the s-word out of me. Are we allowed to cuss? I'll edit it later when I check. Dam can't cuss just checked. Yea dude. It gives you the scariest rush ever, as if you were about to die or something. Plus theres slo-mo. Wicked awesomenessnessness. It's so fun. If you don't play it, I'll beat you up cause you should play it. Theres also a F.E.A.R online multiplayer too. Oh Fword yea!
The Specialist; the most awesomest online fps ever. It kicks ass. You can kung fu,shoot guns, wall jumps, back flips, front flips, awholelotta stunts, AND theirs slo-mo. It's wicked fun. Plus theirs a huge variety of guns and even knives and a katana. Dude if you wanna have fun play this, but you should practice first with bots or else youre gonna get beat up.
Theres my reviews Took me forever to write this. Better be worth it.