Dreamhost uses phpMyAdmin as well. You need to save the databases and data from your old host (EXPORT sql, Structure and data, complete inserts) Save it to an SQL table.
The on the new host create your database, go to the SQL tab, and upload your SQL file from above. Whala! Your tables are in tact.
Moving your files just involves downloading all of them by FTP, and uploading them to the new server. Change the settings for your SQL server, and your site will be up and running.
MP3Mystic Your personal Streaming MP3 Server