You can see what games I have (or have played at least) here. Would like to see your guys' too if you have the links. (it's also an easier way to add people to your friends list, hence why I'm asking for the links)
Ok Now once you connect your router to the back of the xbox go to sign up for Xbox Live. There are 2 versions. Silver (Free, but no online play. Download stuff etc...) Gold let's you play online. It is 60 bucks a year or you can pay Something like 6 bucks a month if you want. Create a games tag ( your profile which will be linked to your Hotmail account). Thats about it I think.
You don't necessarily need a wireless router, there's just a network cable input on the back of the Xbox where you can plug whatever kind of high speed internet connection you have into it.