Hi, after 1 year with Freehostia, I'm moving over to a host with more to offer me.
This means I need to backup a lot of things and reupload them on my new site, however, how can I easily backup my SQL database on Freehostia and save it to a file to later reupload on my other (cPanel) site?

Try this: http://forum.freehostia.com/viewtopic.php?t=240 . Look near the end of the page under the heading : Exporting a database