If you just want to be learning the SQL language, and not database administration, I would recommend working with SQLite. If you're on a Mac, it should already be installed. It is a much simpler system than most RDBMSes; there is no server to set up, and client to connect to the server. There are no directories of cryptic files, or anything of the sort. To get started, you can just type:
sqlite3 mydatabase.db
And start working with it. It's so much lighter weight and easier to set up and use than the other database systems that I think it's a good choice for a beginner.
Now, SQLite is a fairly small and lightweight language. If you need be getting into any kind of really complex queries and data mining, I would recommend PostgreSQL. It has a fairly advanced query optimizer, and a pretty long list of SQL features.
And if you want to learn a database as something to use for back-end storage for web programming or something of the sort, MySQL is what I'd choose. It's ubiquitous, supported by almost any web host, and it's pretty fast for very simple queries and updates, which is generally what you need for a web system. It has some real gotchas to avoid when setting it up; you have to choose between several different storage engines, and it can take a lot of work to convince it to actually work with Unicode data. But it's good to learn mostly for its ubiquity.