Have a backup.

Stop MySQL:

sudo service mysql stop

Move (or copy) your mysql directory. If you want /mnt/mysql to be the new directory, do it like this:

sudo mv /var/lib/mysql /mnt/

Open your MySQL configuration (sudo vim /etc/mysql/my.cnf) and change the datadir value to your new path (e.g. /mnt/mysql)

Modify your AppArmor configuration:

sudo vim /etc/apparmor.d/usr.sbin.mysqld

Change/copy the lines granting access to /var/lib/mysql to your new path. Otherwise MySQL will not start.

Load your new AppArmor configuration:

sudo service apparmor reload

Start MySQL:

sudo service mysql start