Eric, this is really easy, and is exactly why phpMyAdmin was invented.
Dump the schema and the data from your old database. The result should be a big .sql file.
FTP it from your old server to your desktop PC.
Using phpMyAdmin, create the new database on your new ISP.
Click on the new database's name when it shows up in the left rail.
In the main portion of a page you should see a window labeled " Run SQL query/queries on database" followed by the database name. You can cut/paste your SQL into the window right below that ... or, better yet, look right below it, where you're offered the option of uploading. Do that. Slick as a bunny, your SQL is uploaded from your PC and executed, loading your old schema and data into the new db.