First-person shooter would work fine, but i mostly play that and i want to try out new things. Just make any suggestions. If i never played it before i'll give it a try.
Freeware downloads or bought copies? The the former, I'm not really sure - I never paid too much attention to the freeware FPS market, I also am not sure whether you're content with making micro-transactions for better quality in-game items. I can think of Assault Cube (which came up a while back in it's own thread), whilst it doesn't have a ton of features, I think once it gets noticed by the open source community development will boom. It's nice, fast-paced action with both multiplayer and a singleplayer, you vs. bot deathmatch system which prepares you for multiplayer.
I can also think of AA (America's Army). The game is produced by the American Army (which means it lags behind on graphics and features) but it certainly has a lot of attention. It is very realistic and requires so much tactics it makes my brain hurt. It has a multiplayer system that consists of dedicated servers with their own map rotations, etc. and also a single player mode filled with realistic military training environments which you complete to unlock the rights to use certain weapons or other items in multiplayer.
Finally, I can think of Combat Arms which is notorious for the requirement of lots of micro-transactions (small payments for beneficial in-game items, AKA real world trading) and an overload of hackers. I've never played it, but if you get on the right servers at the right time and are willing to pay a few quid (dollars, if you're American) for some items it's great according to some of my mates.
If you're looking for paid games then my first picks would be:
Unreal Tournament - Old now, so you don't need a monster of a computer to run it, but it is faced paced actions, nearly 1000 servers, a large modding community etc.
Doom - it's Doom. Old, so no need for a good PC. Once again, I repeat, it's Doom.
The first four Call of Duties - these are instant classic - MW2 can't even come close to comparing - everyone besides COD4 does not require a new computer.
Medal of Honour: Allied Forces - I thoroughly enjoyed this game, I found the story more immersive than the COD games and also felt a bit more comfortable with it than COD 2 and 3.
Halo: Combat Evolved - the first Halo game. An instant classic and whilst the graphics look dated it is so much fun.
FEAR 1 - this is a horror game that really did make me jump. The shooting is a bit lacking but the game manages to make you scared, it's so fun.
AVP 2010 - I'm getting this. Looks pretty good, but I can't exactly suggest it until I've had a go myself.