As a new user, I would definitely agree that MySQL is the way to go, especially for a web site. Most web servers and Linux OSs include MySQL, and most web programming languages (JAVA, PERL, PHP) either support MySQL by default or can easily have drivers installed to communicate with it. Additionally, most of the tutorials you find on the web about online databases assume that you are using MySQL. As your needs grow and evolve, you will probably look into other databases, but MySQL will probably give you the best starting point.

As for on your (Windows?) desktop computer, I would still suggest using MySQL because it is free, unlike Access, and your interaction with the database will be purely SQL queries without all the vendor specific interfaces that Access provides. You will probably have an easier time learning and familiarizing yourself with SQL using a pure database without the additional distractions - um, I mean features - of Access.