You are going to need to migrate you instalation in two parts.
Part 1 you already eluded to. You will need to copy the files from one server to another. I am guessing you know how to do this so I will not dive any deeper into it. If you do need more explanation, please let me know and I will edit the question.
Part 2 is what you mentioned but said you did not understand. Copying the database of wp install. Wordpress runs off of PHP and MySQL. The "files" part in part 1 is the PHP files (along with some html and css). You need to log into his MySQL server and do an export of his database. You should be able to export the database (How to export mysql database to another computer?) and import it into his new server on GoDaddy. (Error importing SQL dump into MySQL: Unknown database / Can't create database).
Just take things slow, follow the guides that I have linked and do not delete anything from the first server until everything is working on the second. Please let me know if you do not understand anything.