I'm not a 360 expert or anything as I don't own one so I could be missing a step or two but I believe it's fairly straight forward.
You'll need: An Internet Survice Provider, a way to connect your Xbox 360 to your modem & an Xbox Live account.
The price of an Xbox Live account varies depending on the length of of time you'd life to purchase. ($7.99 for a monthly membership, $19.99 for 3 months, $49.99 for one year)
Connect your Xbox 360 to your internet by using an ethernet cable or a wireless connection, and then join Xbox Live & create a profile.
Again, I'm not an Xbox user so this might not be 100% accurate, but I believe this is how it's done.
Good luck & enjoy your games.