Yes I would move the domain. Much easier if you can administer everything in one place. GoDaddy has a decent online help system that will walk you thorugh the how to.
For the switch, I would inform your users that it will happen starting day/time of your choice. I would let them know that the board will stay up but any posts made after that might be lost.
Take a snapshot and move it over to the new server.
Once installed, flip the DNS servers. It will take a little time. Most of the time ist 1-2h but it can take up to 24 (official SLA). I opt o do this kind of work in the middle of the night when there are little to no users on the forum.
Next morning they all wake up to a super fast forum.

- at least in theory. Might want to do some reading up front and print out the articles on how to setup the domains and name servers.
DO a dry run by taking a snapshot of your site and install it on the new server. You can always delete it and start over. Just to make sure you don't run into unexpected issues. And as always: backup, backup, backup before making any changes.
Hope this helps!