Just download the vbulletin your now using to your harddisk upload the whole thing on your new webhosting. Download the database from your phpmyadmin and upload the whole thing in the new hostings phpmyadmin (could be that your database is too big so you may need to slice the thing) after this is done change the config.php file too your new database login and password (databasename).
But don't forget before you upload your database too your new hosting open the .sql and change ALL instances that link back to your old url... and change ALL of them to your new url link.