Moving is fairly straightforward, so I can give a brief rundown on the steps.
1) Close Forum
2) Zip up entire Forum directory
3) Get a full dump of your database, through command line or phpmyadmin.
4) Transfer forum zip and database dump to new server.
5) Unzip your forum directory in the proper location on the new server
6) Import your database on the new server
7) Edit includes/config.inc.php. You'll need to edit the database settings, paths and urls throughout that file
8) Login to new forum, go into control panel and clear the cache.
The database portion may be your hardest part since you said you're unfamiliar with it. If you have SSH access to the server, then you can use mysqldump to get a dump. If you only have FTP access, then your host probably provides something like phpmyadmin where you can get a dump. Some hosts will help with this and do the dump for you.
If you get stuck on moving, we do offer a service to do this for you.
As for when to upgrade. It's really a preference thing. I normally always upgrade before moving.
