Finally moved the domain. It was quite a bit of hard work. I have decided to put a list of things I had to do.
1) I had to register the domain.
2) I had to open cPanel and create the database
3) I had to download every file from the old server. This was probably the most time-consuming part.
4) I had to modify the config.php file to change the SQL server hostname.
5) I had to upload all the files to the new server.
This wasn't very time-consuming, because the new server is waaay faster.
7) I had to go to the ACP (administration control panel) and change the cookie settings.
Thanks to TastyUrinalCake and stabbytaco for their support and donations into this forum. (yep, they paid for the DNS name and the web hosting)