As far as I can tell as a non-designer, there seem to be (essentially) two phases to graphic design for the web: wireframes and final design. The first phase is sort of a high volatility phase where one tries out many different approaches, mocking them up with extremely basic wireframes (in something like Balsamiq or even on paper for the best designers), figuring out what the client really wants (they don't know) and figuring out what you want. The second phase is making something nice looking based on the wireframes and other details from the first phase.
These two phases are completely different. They can, and often are, done by completely different people. To be good at either, you need to have spent years developing a good sense of design. However, other than that the skills seem quite different. For the first phase, one might need layout and communication skills. For the second, one might need to know about typography and color theory, maybe be able to draw proficiently, and so on. Often, the second phase includes turning the design into HTML and CSS, but this is rarely a good use of the designer's time, and can be delegated to specialty providers like psd2html.
Without more knowledge about what your background is, and why you want to study graphic design for the web, I think it is difficult for anyone to suggest what you need to study first. However, given how little of the above was really specific to web design, I think you might be better served by a more general design program, which might also be easier to find.