Generally, if you choose a reputable host, they will be able to assist you with the backup and restoration of your MySQL databases and files. Some do, some don't, it really depends on the web host.
If you're simply looking to move the forum, download the base forum files via FTP and then use phpMyAdmin to select all of the tables in your MySQL database and back them up.
So long as your database isn't in the upper 100MB+ range, phpMyAdmin should backup the database rather quickly.
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