A couple of games to consider especially if you do decide to get the full Live service (which I recommend).
CoD 2 could probably be picked up for not much cash now that CoD 3 is out and it is a really fun game for live. The Single player is also very enjoyable on Veteran setting.
CoD 3 is visually more impressive but still has some gameplay issues in multiplayer which might be patched but imo, as of now CoD 2 is overall the better game, paticularly with the price differential.
I have GoW and have played a little of it and it is quite fun, and visually quite impressive, but if you do not play multiplayer, you might as well rent it as it does not take too long to complete the SP campaign.
I can concur that having played the demos of Dead Rising and Rainbow 6 Vegas that these are 2 titles that are really good, especially the R6 Vegas.
If you do get live I would also suggest downloading Texas Hold Em and Uno (unless you buy the Cam attachment which comes with a copy of Uno)
I know that they do not utilize the power of the 360, however they are great fun to play with your friends on Live when you are not in the mood for killing your friends and just want some light fun with a lot of laughs...and definently get the downloadable version of Texas Hold Em as opposed to the WCP version. Gameplay on the WCP one is awful and the sound effects are downright annoying. The only saving grace of that turd is that you can use the cam to import your face to your player and even that is iffy.
I have only played the demo version, but many of my friends that have kids having been having great fun with the Lego StarWars game for the 360 too. If you are a StarWars fan or have kids, that would be a worthy title to check out, at least as a rental. (the Demo of it is probably still available too)
My biggest recommendation though would be to get the Gold Live subscription, and get a friends list built up of people you enjoy playing with so that you don\'t have to play with the arses that make up the general population on live. It really makes the 360 gaming experience so much better, and many of the new games coming out, GoW (with Co-Op through out the full game), R6 Vegas, CoD II and III are really going to be great for playing with friends, not to mention judging from the demo, Lost Planet looks like it will be really good for multi player as well. (you should check out that demo too if it is still available)
Edit: Oops I forgot all about Oblivion. If you don\'t have it for the PC, this is an incredible game for the 360.