I would recommend NOT to start with MySQL. It often behaves very different from most other databases. (This is not an evil plan, just optimizations for special cases in some applications. And annoying in all other applications.) Good documentation.
Oracle is nice, if you have a good DB admin that manages it for you. Everything seems to be documented, even Larry Ellison's coffee mug. You just have to know where to search for the documentation. And Oracle brings TONS of software, including Apache, Perl, Java and every tool you can think of. Bloatware.
MS SQL is easy to set up (just click the "continue" buttons until they disappear ...), but it can be a real PITA when it comes to concurrent access, locking, multiple active statements, and access from non-Windows systems.
PostgreSQL is easy to set up, clean, well documented, and it has a very short gotchas list. (And for me, it feels like Oracle done right.)
SQLite is small, low-fat, and can easily be embedded. But it has a very unusual type system.
- For fun projects as well as for new commercial projects, I would recommend to use PostgreSQL.
- If you want to learn more, learn Oracle, then MS SQL, followed by MySQL and SQLite. (Clean to dirty order.)
- If you have to deal with web hosting, you should learn MySQL (available nearly everywhere, but often in ancient versions) and SQLite (for hosters unwilling to provide a usable database), remembering that they have unusual behaviours.
- If you want to earn money, you should know all of them, including the "light" versions (SQL Express, Personal Oracle). You sould also try to learn DB2 and other RBDMS.
Today I will gladly share my knowledge and experience, for there are no sweeter words than "I told you so". ;-)