The installation of MySQL is straighforward on debian-like systems. Just type:
sudo aptitude install mysql-server mysql-client -y
In the installation process you'll be asked to set up the password for the admin user. Remember this password because you'll need it in a moment. One thing, though, this username must NOT be used by your applications because it has ALL privileges possible, and you don't want your application to have the right to drop other databases installed in the same MySQL. Limited users must be created for ...