Most people say they get a free month when they sign up for xbox 360 for the first time. I was one of those people so I can confirm it. They are two way to do it, online with a PC and with the 360 itself.
go to this link: xbox live signup
.Once you get there you will have options. There is a free Silver version that lets you download demos and DLC. This can be upgraded to gold later so you can play online.
You can also sign up for gold but you will need to use a credit card or use a code you can purchase at a electronics,department store or even 711. To redeem a code there is a little menu on the left of the screen. Click redeem code and click on "redeem code" and it will ask you for the code. Put your code in these and your good to go!
To sign up with your xbox be sure you have your xbox connected to the internet and start it. When you start it go into the menu and choose the "try xbox live" option or "sign up for xbox live"......sorry I forget what it is says exactly. Go thru much of the same process.
You will also need a "microsoft passport" it will tie in things like your msn email and msn screenname.
Comment back if you need anymore help or with more questions.
see you on live!!!!