If your version of mySQL supports the usage of the mysqldump function that would be the best way?.
MySQL :: MySQL 5.1 Reference Manual :: 4.5.4 mysqldump ? A Database Backup Program

You'd require shell access to the server to do that of course.

the actual database tables/files are found in /var/lib/mysql on most systems, There's also a hacky way to transfer mySQL files to a new server provided you have root access to access the /var/lib/mysql directory.