cPanel (or crappy alternative maybe) MAY LOOK DIFFERENT BUT FUNCTIONS SAME.
If you need some help backing up files, here.
Depending on what host you use, you may get a backup option, likewise here.
If not, then you can always open the File Manager and
Highlight over everything, compress. Download it by double clicking.
If you just use a FTP client like FileZilla, you SHOULD be able to do the same thing, just would take longer.
As for your database, open up PHPMYADMIN,
locate your database, and
Export it. You can then go to the new host, and simply import it using the tab in the above image.
When you get to your new host and move everything in, MAKE SURE to edit your inc/config though, so it works!