Step 1
First setup your drive (assumed it's mounted and setup correctly in the fstab) and create new MySQL directory:
sudo mkdir /path/to/new/mysql
Make sure this folder has the right permissions:
chown -R mysql:mysql /data/mysql
Now here you can shutdown the service while copying data. Copy the data from the current data directory to new data directory, example:.
sudo cp -R /var/lib/mysql/* /path/to/new/mysql/
or you can copy individually each databases if you prefer or if you feel safe, you can move the folder using the mv command.
Step 2
You can change the location of your MySQL data in your my.cnf. If you don't know where this file is, you can run the command:
locate my.cnf
This will give you the path of where the my.cnf is located. Then edit the file:
sudo vi /etc/mysql/my.cnf (of course you have to replace with your path)
Now look for datadir = /var/lib/mysql and update to datadir = /path/to/new/mysql.
Then restart mysql
sudo /etc/init.d/mysqld restart
After few days/weeks, if you feel comfortable you can remove the old folder (/var/lib/mysql) or backup everything inside.