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The root password for for MySQL is not the same as the root password for the server itself. It sounds like you are probably using one of the one-click apps if there is a password set for MySQL. Which droplet are you using?
With the LAMP and LEMP droplets when you first log in, the MySQL root password is displayed in the MOTD (aka message of the day). That is the text that is first shown in the console.
With the Wordpress droplet, you can find the MySQL root password by running: <pre> cat /root/.my.cnf </pre>