1. You're right with copying all files from the current host to the new one. Then you get a identical fileset.
When you have identical files for a forum migration, you only need to change the domain and cookies settings afterwards, mentioned by Matt.
2. The current database needs to be exported (in a solid state, therefore close the current forum before export).
Database export can be achieved by a "Database Backup" from ACP - or any other DB tools provided by your current webhoster. Be sure all tables (with schema and content) are exported.
After creating an export, you can transfer this export to your new webserver and run a database import.
When fileset and database is in place on your new hoster, you can open ACP again and correct domain/URL as well as cookie settings.
Thats all for a forum migration. Good luck!
[ExiTuS]