For a serious site (one want to work reliably, and respond to changes) you should dedicate a droplet to run your mysql server, and a separate droplet for your php+http server (apache or nginx).

On your mysql VPS, for mysql to work “well” (without weird crashes) you need about 2GB Ram, and multiple CPU’s are ideal anyway. That corresponds to the $20/m droplet.

For the php/web server droplet, you may be able to achieve good enough performance with a 512MB ($5/m droplet) if you run nginx+fpm. If you run apache, you should start at the 1GB/$10/m droplet… but you may need a 2+ core system (the ones starting at $20/m) to get decent results for non-trivial PHP based site under any kind of load (on apache).

My recommended starting configuration (without getting into replication, which you should consider if your data is important):
$10/m Droplet running Ubuntu 14 LTS + php5.5 + nginx - fpm
$20/m Droplet running Ubuntu 14 LTS + mariadb-server (or mysql-server)

Remember, most of the DigitalOcean tutorials assume you run mysql on the same droplet as your web server. Those are ok to use, just SKIP the mysql part.

To configure Ubuntu 14 LTS + php5.5 + nginx - fpm: (SKIP the mysql section)
https://www.digitalocean.com/community/tutorials/how-to-install-linux-nginx-mysql-php-lemp-stack-on-ubuntu-12-04

To configure Ubuntu 14 LTS + mariadb-server (mariadb is my preferred fork of MySQL)

1. create new droplet with Ubuntu 14 LTS
2. from command line: apt-get install mariadb-server