Recently, I had to deal with migrating a website (not one of mine) to a new host, including an installation of the content management system Mambo. Although not a major hassle, I don't normally deal with this sort of thing.
I'm sure there are easier ways to migrate Mambo, but here is how I did it:
1) Made a partial backup of all types from the previous host in Cpanel > Backups. Of interest to Mambo were the home directory, the databases and the email forwarding.
2) I had issues with restoring the home directory backup on the new host, so I uploaded it in Cpanel > File Manager on the new host and unzipped it on the server, moving all files and folders to the public www directory (that didn't already exist there).
3) I uploaded my old Mambo database using the "restore partial backup" option in Cpanel > Backups.
4) I checked configuration.php in Cpanel > File Manager, and added a new, identical user name and password in Cpanel > MySQL Databases. Then I simply added that user to the Mambo database that I uploaded in step 3.