I decided to install and try MySql. The process was quite painless. I installed it on Windows and chose the setting to run it as a service which boots with the OS. However, when it came time to create tables and view their contents, it was quite shocking to me how primitive they are. Coming from a Microsoft background where the tools for Access and Sql Server are excellent, it's really stone age time with MySQL.
I've heard that PostgreSQL is a better database, but I'm not too concerned for now, since my needs are quite simple. A quick perusal of the site showed that there aren't really any better tools (at least free ones) than those available for MySQL, so there's no pressing need to change. It will be more important to choose a database that works well with Hibernate or Ruby on Rails, but that's a decision left for later.