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RPGs, Action Games, Sports, Family/Casual games, and those Kinect games.
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Seeing as how you probably don't want to spend too much cash in one go, try these on for size:
Rpg - Last Remnant. Sure it has graphical issues and sure most of what you need to know to play the game well isn't explained anywhere in the game but it's a proper time sink of a game if you get into it. Also has the rare distinction of being a jrpg where you can actually lose battles and not just be able to grind your way past the tough bits.
Action Games - You really should buy Halo Reach, or even Halo 3. Both have a decent enough single player campaign and you can of course play co-op. Failing that, get Earth Defence Force Insect Armaggedon. It's silly, it's fun, it's multiplayer. Also the Battlefield series is great. Bad Company 2 had a playable single player campaign and there is of course BF3 coming out very soon.
Sports - GRID is the best driving game on the 360, at least in terms of the fun, if not hyper-realism. The Tiger Woods games are all pretty good and all if you like golf.
Family/Casual - The Lego games are pretty awesome. The Guitar Hero games are top fun too and you can get all the bits dead cheap these days too.
Others - I don't have Kinect so can't comment on that but you should get Condemned, Just Cause 2, GTA IV and more importantly the episodes, Crackdown, Batman, L.A. Noire and Too Human...
Jam it back in, in the dark.