I often look at the Raspberry Pi with two use-cases in mind. An always on / convenient place where I can experiment, perhaps do a little learning or development, and as a learning tool for teenagers to play and potentially learn real world programming and development skills. Although it is absolutely possible to run Mysql for example for database development, the more applications you run the more strain is put on the limited resources of the Raspberry Pi.
SQLite is a software library that implements a self-contained, serverless, zero-configuration, transactional SQL database engine. Crucially SQLite is used in many real world applications, and a whole host of development frameworks and languages have direct support for it. There are many tools to help, even an extension for Firefox that helps to manage SQLite databases. In most cases something you develop with SQLite can be quickly converted to using Mysql if needed later.
SQLite is very portable, the database is a single file and you can do things like working on a database on your Raspberry Pi, moving it to your phone and scripting something to read the data there.
The SQL and database skills learnt in SQLite are transferable to working with other databases.
Best of all SQLite is probably already installed on your Raspberry Pi, it will be already there if you are using the Raspbian image for example. So all you need to do is start reading the documentation.