If you have CPanel support, find and click on the link that says "Backups" then you will see a label "Download a MySQL Database Backup" and bellow that all your databases. Click on the one you want to download on your PC. After that, go to your new cpanel from the second hoster, click again on "Backups" find the label that says "Restore a MySQL Database", click Browse, select the database sql from your PC and click on the Upload button.
If you wan to do it with phpMyAdmin, then:
1. log-in to phpMyAdmin
2. find and click on the link called 'Export"
3. Chose the database name you want to backup and check the Save as File option
4. Click Go and download it to your Desktop
5. Go to your new phpMyAdmin hosting account, log-in and click Import
6. Chose the sql file from your desktop and click Go.
That should do the trick
