In addition to the HTML Level 2, the things I think any budding Web designer would want to work on next are:
These will give you a good basic toolkit to get started. You can create basic websites fairly efficiently with such software and skills. If you want to go further, though, you have to go a little deeper. And you will find that you need to specialize. Creating a dynamic Web site, with database integration, e-commerce, server security, professional graphic design, interactivity, etc., in other words, the kind of site a major corporation would want to create and maintain, requires a team of specialists. I happen to think that this specialization is where the major action is. The single-person all-in-one shop is being relegated to the fringes, and unless you want to be on those fringes, you need to start thinking about where your talent lies.
For instance, if you are most interested in the visual aspects, in how things *look*, you would want to start looking at courses in graphic design, color theory, typography, etc. You may not find these online. Perhaps a local college will be the best place. Many are now creating degree programs in areas like this.
If you enjoy digging deep into the details of code, you might want to go further into programming, and add things like JavaScript, Java, and Perl to your repertoire. There is a lot of demand for people there. Some of these courses are available online through SmartPlanet, though you might want to also look into college courses to supplement what you find online.
If you want to make a lot of money, learn databases and how to integrate them into Web sites. Six-figure salary for sure if you know your stuff. Again, this requires a fairly detail-oriented person, similar to the programming type of person.
If you found yourself attracted to things like the discussion of FTP and server issues, you might want to look into server administration. You have to be a bit of a computer geek for this one. If you have a problem with your computer and go looking for someone to fix it, you are not the kind of person for this job. But if, when you have a problem, you take out the manual and start reading it, thinking "I've got to be able to figure this out, it can't be that hard," you are exactly the sort of person who would do well here. By the way, if you wonder how an economics professor (me) got into teaching computers and such, it is because I always got out the manuals and figured things out. ;)
If you always thought that being a Librarian was an interesting job, and categorizing and organizing information is appealing, you might want to look into Information Architecture. A way to test this is to ask yourself if you really found the discussion about whether something was a header or a paragraph quite fascinating. An Information Architect needs to think about stuff like that all of the time. This is the sort of person who can design an XML application that makes sense.
Another useful specialty is User Interface and Usability. If the idea of studying how people use a Web site, observing which buttons they click on and why, collecting data on who visits a site and what they do when they get there, etc. appeals to you, this is an area to consider.
These are the major directions you can go in if you want to be a part of one of the teams that design the major sites out there. Of course, you may find that none of these things appeal to you. In that case, you are looking at that "fringe" area, which may be just fine as long as you know what you are getting in to. You aren't going to making the big bucks or working on the major sites, but you can do tolerably well. You will be working as a self-employed small business person, and will need to work hard to find clients, and will have all of the headaches that any small-business owner has to face. Maybe you will do this a part-time occupation to supplement your income. Or maybe it will be a hobby that never earns much money, but lets you help out as a volunteer for a church, a school, or a charity. In that case, basics like the CSS and graphics, and maybe a little JavaScript, are still the best preparation for such a goal.