Both Oracle and IBM offer offer versions of their flagship databases for free in a similar manner to SQL Server Express. Both are excellent products if you wish to play with a 'real' database system.
Oracle 10g Express
IBM DB2 Express
Postgres is also well worth considering as it is also a 'real' relational database, having originally started as the much respected Ingres RDBMS in the 1980s. However Postgress until recently was Unix-based and I've never found it particularly reliable installing on Windows.
Personally I would not recommend MySQL for your scenario - although the latest edition can finally be described as a RDBMS having evolved from what really was a collection of flat files with a limited SQL query interface it's peculiarities probably mitigate against it's use as a learning tool unless you are actually focused on MySQL. To placate all those thinking of giving me a down vote for this - yes I do use MySQL, and yes it is (now) and excellent product in its place - just not here.
There are a number of other less popular RDMSs you could consider. Firebird has always been something of a favourite of mine as I used to code a lot of Delphi, and SQLite is a joy to work with in many circumstances. It's also a pity you cannot use SQL Server Express as the SQL Server product range is simply the most solid product Microsoft produces and usually my database of choice on anything but the smallest, largest or most unsual installations