Get a complete copy of the DB using phpMyAdmin. Get a complete copy of the SMF directory. Put the old SMF directory onto the new server. Create a database with the same name. Use phpMyAdmin to import the old database into the new one. That should work, but make sure the new and old SMF both have the same path (eg. yourdomain.com/forum and /public_html/forum)
If PhpMyAdmin is taking too long to import the database, you can upload the .sql file through FTP and import the database through command line. mysql -p -h -f locahost NEWDATABASENAME < backup.sql If you don't have SSH access, most hosts will do this for you.