If you're looking at a scalable open source choice, I would go for
PostgreSQL, which IMHO is a more robust alternative than MySQL. Plus I like elephants.
If you want to try a commercial vendor, you can start with the free
Oracle Express Edition (Oracle XE), which has a restricted licence but can be used commercially as well as for personal use:
Oracle Database XE can be installed on any size host machine with any number of CPUs (one database per machine), but XE will store up to 11GB of user data, use up to 1GB of memory, and use one CPU on the host machine.
Oracle XE is fully compatible with enterprise Oracle, so you can migrate your applications easily to scale up in future. Oracle XE includes most of the core RDBMS tools you will find on any Oracle edition, including PL/SQL and a load of other stuff. I often use Oracle XE to prototype applications that will eventually be implemented on an enterprise Oracle platform, for example.
Combine Oracle XE with the free
SQLDeveloper tool, which includes data modelling options and can also be integrated with version control tools like Subversion, and you have a pretty good basis for database application development even before you start building your
Java stack. Oracle XE also includes the free
Oracle Application Express (APEX) tool for building data-driven web applications really easily without Java, but I'm guessing nobody around here would talk to you if you used that!
There is also a
free edition of Microsoft's SQL Server, which is probably easy to integrate with Visual Studio and .NET etc, but I don't know anything about it. Maybe I'll try it out...
All of the above options are serious relational databases that will give you access to many of the core features available on any enterprise RDBMS, so any work you do on these will teach you useful skills for working with any database.
I wouldn't use Microsoft Access.