Congrats on the purchase, Josh. Despite being a Playstation 3 fan and supporter, I'm not going to lie and say the Xbox 360 isn't a fine piece of work.
As for recommendations;
I, personally, wouldn't recommend Halo 3: ODST. While many people think it's a blast, I personally thought it was rather boring and had a very low fun factor. The original Halo 3, on the other hand, is a blast. Played through the entire campaign with a friend the day I purchased the game. It's still fun when you're on solo, but if you want to get the full experience out of this title I'd recommend getting a bud to come over when you play it and do some cooperative.
Another title I'd recommend would be Just Cause. It's an older title, and somewhat outdated, but it's something that even today I still play. To be honest, the story [in my opinion] was not very strong, however, the environment and the ability to just free roam around the setting is more fun than you'd initially think. Quite frankly, I went through the campaign of this game just so that I could get the full experience out of the free roam.
For my last suggestion, I'd say Crackdown. It's another older 360 title, yes, however it's another one that got me hooked. With Crackdown 2 on its way, I'd recommend purchasing this title if you can find it cheap, and taking a good, solid run through it. Once you get used to it and get past some of the first missions, I think you'll get into the game very much like I did when I played it.
Hope these suggestions help you select titles in the future.