At your old host, access phpMyAdmin, you're going to backup your database(s) which will ultimately be downloaded to your computer.

Click the Export tab and a screen similar to the following should be displayed (click for larger image).

As per the image above, select all tables and check the entries as indicated. Make sure to check "save as file" and select compression mode required.

At this point click the "Go" button and your download manager or save as file dialog should appear. Under my system (firefox) the following appears.

Accept the file, click 'Ok' and an sql file will be created and downloaded to your computer. The file downloaded, is for all intents and purposes, a backup file of your database. This file will be imported into a newly created database at your new host.