There are many ways to do few things but here is one.
Firstly download and backup your site files from the other hosting account. Do this before cancelling that hosting account. Important!
Go into cPanel and 'manage hosting', look for the IP address for that hosting. When you have done this go back into 'manage DNS' settings (don't change default name servers) and forward to the IP address of your new godaddy hosting. Then using your backup, repopulate your new hosting.
Bear in mind that services and widgets or programs and extras that worked on the other hosting providers servers may not be entirely compatible with your hosting package with Godaddy. When I say repopulate your hosting I mean it in a general sense in that most of your work will be saved or usable if a similar hosting package has been chosen. Different hosting providers vary in what they can offer or support.