Other people have mentioned a lot of the essential purchases, so can I mention a few of my favourites from XBLA?
Even if you don't have any points, you can still get a few things completely free. Such as Hexic HD, Doritos Crash Course, and Harm's Way. (Does anybody know of any other free games on Live?)
If you like puzzle and strategy, then I can recommend Catan. It's a great little game which supports online play, and I find myself playing it most nights. I hear Carcassonne is also a good board-game.
And finally, don't forget to look at the Indie Games section for some incredibly cheap minigames (although the quality varies wildly). From this ection I can recommend Miner Dig Deep for some serious OCD-soothing times, and I MAED A GAME WITH ZOMBI3S INIT which is a fun little twin-stick shooter. Remember all Indie games have trial versions so you can play before you buy.
Welcome to 360!