MySQL is one of the most popular database engines on
the market, available in a
Community Edition which is free of charge.
It provides the same features Microsoft SQL Server,
Oracle, and DB2 offer for the most common database needs
(in other words, you aren't likely to need to pay for
one of these more expensive database engines unless
you have very specialized, high scalability/performance/availability
requirements). Best of all, services like Amazon
RDS offer inexpensive MySQL hosting at a low monthly
price. You'll also find most internet service
providers that offer website hosting will include MySQL
databases as part of your package free of charge.
Finally, if you are familiar with MS Access query language,
you'll be well prepared for working with MySQL views
(essentially the same SQL concept).