I just got an XBOX 360 Arcade edition as a gift from my brother and I'm very new to next gen systems and am without an HD tv/monitor. Can anyone recommend a few good/must have games to start out with for sports, first person shooters, flight/air combat sims and of course Star Wars?
Be sure to get you Live set up so we can play sometime.

As for my game recommendations I differ from Matt and Jesse in that I really prefer the Halo series over the COD series in every mode, both single and multiplayer. There's a lot more to do, and it is much easier to play with more people. When playing online COD only lets one person play, for Halo you can play online with guests on your Xbox. I think Halo is a much better game. CoD is still worth picking up though, I'm not saying that the games are bad, Halo is just better. I could go into more detail on that if you'd like me to, but I'll just leave it at that for now.
The Rainbow 6 Vegas games are good for single player. I never play them on MP often enough to give an educated opinion on that aspect. I think they are greatest hits so they may be cheap now too.
HALO3 is good but I've never been a diehard HALO fan for the SP games. I like the mulitiplayer for it better. HALO Wars is out too but I've never played it. I'm pretty sure it's just basically new maps and stuff for the HALO series, which is a good thing for multiplayer.
Halo Wars isn't a FPS at all. If you bought the Special Edition you could It's an RTS (Real Time strategy for those of you who are unaware what that means) that is set within the Halo Universe. It is a little different trying to get used to an RTS on a console, but they actually made it work pretty well. The controls are nice and streamlined and they managed to fit a lot of functions on the controller without leaving you overwhelmed by the controls. Jesse, I'm actually kind of surprised that you aren't more into the Halo campaign/story/universe. It seems like it would be right up your alley. It's a really detailed story with a fully fleshed out and functional and fairly realistic military aspect to it.
The Rainbow 6 Vegas games are good for single player. I never play them on MP often enough to give an educated opinion on that aspect. I think they are greatest hits so they may be cheap now too.
Force Unleashed is definitely worth picking up. It's a fun game for sure, and I really enjoyed the story. The controls can get a bit frustrating from time to time and the hardest difficulty is damned near impossible (I don't know how Robb beat it.) I think I'm in the minority here but I hated the LEGO Star Wars game. It was kind of fun/cute for the first few times I played it, then I thought it was just too repetitive and beating it 100% took FOREVER. But it should be $20 or cheaper if you find it used so it isn't let it sets you back a lot of cash if you don't end up liking it.
One of my favorite games I've been playing a lot lately is Mirror's Edge. Visually it is a beautiful game, the world has a very unique look to it, and it will really make you want to pick up an HD TV. It is a first-person action game. There is some fighting but that isn't at all the core of the gameplay. It's a freerunning game and it is a very out of the box game. It has a simple and fluid control scheme, that is really easy to pick up on. And it is has a little bit of the puzzle aspect to it, trying to figure out the best routes to take. Plus the story is pretty neat too. I can't recommend it enough. (It is single player only BTW)
Another classic that should be cheap and is one of my all time favorites is Assassin's Creed. I think some people may have felt it can get repetitive, but i can get enough of the game. The world is gigantic and there are some stunning visuals. It's another Single player only game but I simply can't get enough of it. I'm so psyched for the sequel, everything they have been showing at E3 has me really excited (this years E3 has given me a full on video game hard on all week, there is so much stuff in the coming year to be excited for.)
Mass Effect is an absolute must have and is also one of my all time favorite games. It is made by BioWare, the company who made KOTOR 1 and is making the new SWTOR game. The game combines the best elements of a sandbox RPG with FPS/3rd Person shooter elements. It's very much an interactive cinematic experience. It's also a single player only game.
The Gears of War series is also very good. I personally like the single player better than the head to head Mulitplayer. It does have online Co-op which is always fun. The Horde mode on Gears 2 is great to play with a few friends.
If you are interested in stealth combat. Tom Clancy's Splinter Cell is the way to go. I really enjoy those games. (They've only made one for the 360 so far, though the next one comes out this holiday season.) It has multiplayer, but I don't think too many people are ever online. I definitely like it more for the campaign than the MP.
I haven't played Left 4 Dead, but I have heard it is fun. Though I'd wait on buying that one. It just came out a few months ago and they already have the sequel coming out within a few months. So there won't be any DLC for the first one, and it will probably drop in price when the new one hits. Or you could just pick up the second one.
As for Sports games it kind of depends on what you are into. I generally suck at them so I don't play them often but if you are looking for a Basketball game NBA 2k9 (by SEGA) is better than NBA '09. (by EA) Generally though EA Sports makes the best sports games. Except for Madden '07 and '08 which flat out sucked, don't waste money on them. I have only played '09 a little but it seems like they were able to fix '07 and '08's problems and it appears to be a much better game.
Like Matt was saying you'll want to pick up a hard drive at some point, it is fairly essential to have. Even if you go for a smaller size. I have the 120 GB and I only have about 10-15 GB left on it.