For obvious reasons I'm going to steer clear of the commercial games and give you a taste of a few open source games and what better one to kick things off than Nexuiz!
Nexuiz is a very fast passed FPS game with bot and multi player support. The bots are a huge plus since not all people like to play online (like me) and they are very clever and at higher levels can give you quite a nasty hiding!
The game is a mix between Quake3 and UT2004 with the gfx a bit more on the Quake side (not as good as UT2004) and the gameplay is explosive!
You run around the map collecting guns and munition and frag the one that comes in sight, but imagine Quake3 at twice the pace...
You have your campaign mode (more like a "see how far up the ladder you can go") and the usual instant action (bots) and multiplayer options.
Once you played this and head on out for a quick UT2004 death match it will feel like you are playing a RTS instead of a FPS. At first this will probably frustrate the n00bs, but as you skill improve it is non-stop trigger happy action.
The trick is to time your action so that you can get to a weapon, because see, once you picked up a weapon it disappears from the map for a while (like the ammo, adrenalin, etc. on UT2004). So if you are not fast of the trigger and cannot find a weapon fast, you are pretty much cannon fodder!
Once I got the feel for the game I swore off the sissy games like CS and UT2004. This will keep you busy and well entertained for quite a while!