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).
There you go. Quality games I'd choose from.