Its not all that complicated. First you will need to create a mysql database on your DH account let's call it MyDB1 (just an example) and add your user and password for that database. You can either use an existing hostname or create a new one if you haven't already (mydb1.mysite.com for example). Next you will need to export the database on the old host via phpmyadmin (or whatever they have for mysql administration at your old host). Then you will need to go to phpmyadmin link that's beside the hostname you chose (or any of them really since if you already have some they would be interchangeable). Log in with whatever user and password you created for the database you want to use. Once you are logged in just click on your database name on the left (if there is more than one). Click Import on the tab at the top, browse to your sql file (that you exported in a similar fashion on your old host) and then hit go and you should be gravy.
When you are done with that it is just a matter of altering any configuration files on your website that were pointing to the old database and put in the information for the new database, hostname, user, password and the same for anything referencing your old domain name you would need to update it to the new one.
If you still can't figure it out I'd be more than happy to do it for you for a fee (just like everybody else lurking on here would I imagine).