Define "good designer." I'm serious, one of my daily tasks is taking fixed pixel designs - and most of them are beautiful - and trying to make working sites out of them. As a brochure, nice, as a web site, not so good but they do it anyway because for most clients and "designers," it's like art: "I don't know art, I know what I like."
Anyway, for the original question, take this constructively because that's how I intend it:
I have been designing websites since 2004 and self-taught myself HTML and Dreamweaver.
These are not strong skills. There are far more factors involved than just plunking together pages. You can, however, narrow your niche and work **just** with design: providing .PSD's for developers to break up.
The best advice I can give at this point: Get on Elance for a few months and pull down some jobs. You won't make a lot of money, but the experience will give you a sharp insight as to what you're up against in the "web design world." It's pretty brutal, you'll be competing against those working for supplemental income (AKA beer money) and those to whom $50 will feed their family for a week, and working *for* people who want a mountain of work for a pittance - i.e., the average "web design client." You will stumble across a few good ones though, who respect what it takes to do what we do.
Only those few nuggets and your dedication and love for what you will make you stand out and survive it. It's really too much for many people.
edit: I'll add, in respect to "self taught" in the areas of client or server side programming, you *can* do this but you shouldn't. Too many people already do, and here is where "a little bit of knowledge is dangerous" rings true. When you start interacting with a server, you open the potential for security issues. Take deep classes in programming and pay strong attention to security if this is something you want to get into.
[edited by: rocknbil at 5:00 pm (utc) on Sep 8, 2011]