FPS: Team Fortress 2. Completely free, and not as hardcore as some other FPS's out there. I play it pretty casually, and I don't feel bad about that since I didn't put any money into getting it at all.
Single-Player: Borderlands - very fun FPS. Most FPS single-player games are kind of bad in my opinion, but I really liked this one. I like Fallout as well, but something about Borderlands is just better for me. I love the art style as well. Very unique.
Mass Effect - (1 and 2) My two favorite games of all time. ME2 is significantly better than ME1 gameplay-wise, but both have great storylines. I love these games to death, and the replay value is very high. The story itself can take a long time depending on how many of the side-missions you choose to do so you won't have to worry about it being too short. While I wouldn't call it a traditional RPG, if you enjoy RPGs you'll probably love these games.
Minecraft - More fun if you have a friend with a server, but singleplayer can still be very fun too. Not for everybody though. I know some people who really can't get into it simply because they lack the creativity or motivation to create anything for themselves, so it might not be your cup of tea. It's certainly worth a try though, and it's fairly cheap, so I'd recommend it.
Darksiders - Very great apocalypse game where you play as one of the Four Horsemen. The replay value isn't great, but the story is fantastic, and the gameplay is just awesome. And with the sequel coming out in the future (which also looks awesome), now would be a good time to try it and see if you want to buy the second one too. If you love beating the shit out of things with huge swords, this is a goodie.