I'm not a professional web designer/developer just yet.
I got my BFA in Painting & Drawing from
this place, but, since web-related stuff is where I feel like heading in terms of a career, I sometimes wish that I had done graphic design instead.
However, I am one who believes that graphic design
is something that can be learned (though talent makes it easier).
Like cam said, if you want to do development (code) then learn things like PHP, MySQL, XHTML/CSS, JavaScript, and the rest.
If you want to be a designer, take graphic design courses and study interactivity - especially HCI (human-computer interaction), which I took a course in and found to be revelatory. Learn tight XHTML and master CSS.
I'm a firm believer in the idea that what it says on your degree doesn't mean squat. If you're motivated to learn yourself, take some classes in the basics (if you like), read a lot of books - you'll be just as well educated as anyone else.
I still need to learn all the other things cam mentioned - versioning, working in a team, and so on.