You would probably need to check that your webspace has MySQL capabilities. I can install MySQL but you need access to the actual server (which most web hosting won't let you have).
Moving a PHP forum can be quite easy, the worst part is the download and re-store of the database. When I switched SWB over to a new server I split the database back up into about 20 different files. Most of them uploaded ok, but the larger ones needed to be opened in note pad and uploaded in about 50 stages.
Regarding the actual PHP files, the only thing you would need to change would be the username and password in the config.php file to match the new MySQL server.
If your new web space doesn't have MySQL it may be worth buying new web space.
I use
http://www.totalchoicehosting.com for SWB + 2 other websites, and also 6 websites for work and they never really give me much problems.
The other thing to check is the domain name to make sure that you can change the name servers over to match the new domain.