Note that the Wireless N adapter for the 360 will only be able to realize its maximum potential if you are also using a Wireless N router.  If you aren't, you're better off getting the original one.  The N adapter will still work with a G router, but you will only get the speed and range of the router you have.

Personally, I use a wired connection, because my Xbox is about five feet from my router.  Depending on your set-up, that's all you'll need too.  For the wired connection you don't need to change anything on your XBox's network settings.  The console should automatically detect that there is a connection to the internet. 

I'm not sure about the set-up for the wireless adapter, but I can't imagine it's much different from setting up a wireless connection on a PC.  Go to network settings and choose your network, then enter the security code or password or whatever you've got for security key on your router (hopefully you've got something!) and then you should automatically connect anytime you turn on your 360 and sign in after that point - providing, of course, the router is actually connected.

JB