Instructions
Navigate to your Xbox 360's dashboard. (See Reference 2.)
Select the option to sign up for a new Xbox LIVE account.
Fill out the questionnaire. It will ask you for your name, email address, street address, and other contact information. Once all of the information has been entered, continue to the payment screen.
Select a Silver Account and confirm that that is indeed what you want. It will notify on the screen that as a new user you are eligible for a free 30-day trial of a Gold Account.
Agree to accept the free 30-day trial and enjoy.