Probably the best way to proceed is to read the instructions on the moniker website and then contact customer support for any parts that you do not understand. Otherwise, here's a somewhat abbreviated version of the procedure:
At godaddy, for the domains you want to transfer to moniker, you will need to unlock them and also get their authorization codes. Explore around a bit in godaddy and you will find options to perform an unlock and acquire the associated authorization code.
At moniker, log in to your account, and select "transfer domains" from the top of the control panel. The next page tells a little about domain transfers; click on the button under "Transfer a domain" or "Transfer multiple domains now". Both buttons go to the same place: a dialog box requesting the domain name and the authorization code (the one you retrieved earlier for the associated domain at godaddy). The format is: domain name comma authorization code (for example: cnn.com,ae34rfdlkdf). At the bottom of that page, there's a button for "Transfer". Click it, and you should be taken to a billing page to add another year to the domains' registration (a typical part of the transfer process), and once you've paid, all the rest of the instructions will come via email. If you do not receive any emails, be sure to check in your spam folders. The emails will generally be links to approving or canceling the transaction.
The transfer process generally takes a week or more -- moniker will provide a daily status email.
