The 360 really does service a lot of different tastes, the shooters just get the most prominence.
My play time for the 360 has dropped off pretty dramatically after 2008 though. I can't stand the NXE.
Here's some retail games I'd recommend:
Beautiful Katamari - Great, quirky game. Lots of replay value.
BioShock - Incredibly atmospheric shooter. Excellent references to literature.
Condemned: Criminal Origins - This is a really fun game. The weapon variety is a blast. It's only good for one play through, but this is fine since it's ridiculously cheap.
Crackdown - This is an excellent open world game. It's really fun to level your character and collect orbs. It's a blast to get online and play with friends.
Forza series - Incredible racing games. I think that's enough said there.
The Godfather: The Game - A pretty underrated open world game. This was the first free roamer that I actually attempted getting 100% on. Never happened
(damn safes)
Halo: Reach - I really didn't care for Halo 3. It felt... wrong to me for some reason. Reach is really the whole package, though. Everything you could want in a game is here.
Infinite Undiscovery - An action RPG game. It starts off slow and bland, but it really gets going later on in the game.
Mass Effect - Maybe it was just me, but this was the kind of game I'd always wanted to play. There's a lot of exploration to be done, and the conversation system is really engrossing.
Ninja Gaiden 2 - There was a lot of criticism leveled at this for becoming a... well, level based game, compared to the original where levels where somewhat linked. I think it streamlines the game a lot and makes it far more enjoyable.
The Orange Box - An excellent deal, and it has more of the Half-Life you've been enjoying.
Prince of Persia - It's short, but I had a great time with it. It's an interesting game, the art style is great.
Project Gotham Racing 3 - I don't know if I've ever played a game more than this one. The arcade/realism blend here is perfect.
Silent Hill: Homecoming - This game was another one that got some harsh criticism. I can't figure out for the life of me why, though. It's an excellent little horror game.
Stranglehold - This is like Max Payne turned up to 11 with the story shaved out. Don't go in expecting a lot, and you'll have a blast.
Super Street Fighter IV - IV isn't my favorite game in the series, but it's still great to play with friends.
Table Tennis - A surprisingly deep game. This is another one that should be cheap. It's very addictive, also.
Tom Clancy's Ghost Recon Advanced Warfighter - I was surprised at just how fun this game was. I didn't care at all for the earlier Ghost Recon games, but something about this one just clicked with me. Same goes for Rainbow Six Vegas.
Tom Clancy's Splinter Cell Conviction - Really short, but the price has dropped pretty fast. The game moves very fast and is really cinematic, which makes for good replay value.
Downloadable games:
Banjo-Kazooie - It's a classic game, and having the graphics nice and upscaled is a blast.
Pac-Man Championship Edition - I've always enjoyed Pac-Man but... you know, it's old. It's still a classic, but you know what to expect. Championship Edition has the first new maps they've made for a long time, and they're excellent.
Penny-Arcade Adventures Episode One - An excellent and humorous RPG game. It goes fast, but if you're a Penny-Arcade fan, it's more than worth it.
Scott Pilgrim Vs. The World: The Game - This is a truly excellent beat 'em up style game. The soundtrack is outstanding. The game starts off slow, but it has a leveling system where you unlock new moves.
Super Street Fighter II: Turbo HD Remix - Super Turbo is my favorite version of Street Fighter II, and seeing it with redrawn HD graphics by Udon is a treat.