cPanel --> PHPMyAdmin
1. If you click on 'Export' on the first screen, it will dump all your databases (the screen that comes next allows you to pick which ones). You can have it in text format (can be huge), .zip or .gzip ( roughly 20% the size of plain text) and it downloads directly to your computer. For each database, this includes the database creation definition, which often will causes problems importing to another site.
2. Instead, click on the database name on the left column. Next screen, click on 'Export' at top of screen. This just saves the table definitions and data and is what you want if you are going to import into an empty database.
I usually do #2 and make two dumps, one .zip and one .gzip. You can always unzip them locally and use the resulting text file for import.