A lot of people have offered good reply's but I'll offer my own.

I personally use mySQL and I love it. It meets my needs and it's free. (for personal use/open source software) PostgreSQL is another good database. Either way you are mostly using SQL commands which are pretty much interchangable. The only other database I would suggest you look at would be oracle. While not free I think they have a stripped down version for free.

For learning, either mySQL or PostgreSQL are both good options. Pick on and go from there. Perl is very good at accessing databases. (Oracle, mySQL, PostgreSQL and more)