The first thing you need to clarify is what you mean by 'web design'. You say you enjoy designing the look of websites but don't mention the code or technical aspect much. There is a distinction between a web developer and a web designer, and to be what you say you want to be requires a bit of both.
You say you study art so you obviously have graphic design knowledge. If you're not really thinking you'll be good at the coding and technical aspect of web design, then a good route would be to go into digital design work rather than web design purely. You could look into logo design, print design (brochures, posters, leaflets, learning Adobe Photoshop and Illustrator will be pretty much essential) etc along with purely designing web designs for a developer to then 'develop' into a functioning website. This is possible as a career too, you'd likely work in a design agency. However it's obviously better to be able to do both sides yourself.
If you think you are able to learn HTML & CSS (along with some extras, JavaScript, PHP etc if you want to make a career out of this) then you're likely to be able to do really well. You'll have to explore a lot about design and the general conventions of web design to be able to come up with unique but easy to use designs. You'll then need to translate these designs into code using HTML.
To start learning HTML I recommend WC3 schools. Build yourself a crappy little website about anything. Improve it, make a new one, just fiddle around doing your own thing to get the basics. There are plenty of books out there that will help you too. The best thing I could say is to get into a routine about building up your work from a wireframe to a full design (just like you build up each of your art projects with moodboards, research etc. The same applies to web design). When designing a site I'll start with a rough pencil sketch just scribbled out showing the general areas that I need to create. Websites are generally just collections of boxes of different sizes in terms of layout. Once I have that rough sketch I flesh it out in a design program like photoshop into a full 'comp' or design. Then take that back into Dreamweaver to write it up in a wireframe using divs, and then flesh out with colours, images, and content.
If you have any questions feel free to ask. I studied graphic design at A-Level which is when I decided I wanted to get into web design, like you. I'm studying Multimedia Technology and now I'm working as a client-side developer for a digital marketing company.