Transfer your forum files by FTP.
As for database, if both servers store mySQL databases in the file structure (which they should be if you're on dedicated or VPS) you can transfer it by ftp too. Usually it's in a directory called "bin"; take the entire mysql folder out of the bin directory on your old server and put it in the new one. Update config.php if needed.
If not, you can transfer the database manually by logging into your phpMyadmin on your old server and download your forum database by clicking your database name in the left and then Export along the top.
Upload it to your new host by again going into phpmyadmin and selecting Import on the homepage. You'll need to add a user to the database and again update config.php with this new information.