I have absolutely no formal training whatsoever. And, while I cannot claim to really know very much about what I am doing, the little I do is completely self taught (courtesy of places like this). This suits me very well, as I only dabble for my own pleasure, as well as the occassional bit of freelancing on the side. But, I do not think there are too many companies around who would consider hiring anyone without proper qualifications...
It really depends on how far you want to go.

I'm self-taught and could dash off a few things that might help:

1) If you're just starting out, find a professional designer in the area who will let you shadow him/her for a few months. Make sure it's someone you're comfortable with so you can ask questions. Watch what they do, what software they use and what techniques they employ.

2) Starting buying books on Photoshop, web standards, accessibility and CSS. Check Amazon for rankings -- I'm sure alot of folks here can make recommendations, too.

3) Use a feedreader and subscribe to every high-quality design and development feed out there -- too many to mention. Scan them every day. Watch what the best developers are doing and saying, and view their source code often.

4) Subscribe to feeds from every major CSS gallery. Study how the designs work in conjunction with the CSS & markup. Make your own local copies and practice editing the code. Do pro bono projects for practice.

5) Prepare to do all of this on a daily basis for possibly a couple years, and consider it a "hobby" until you're ready for a full-time position. Then start putting together examples for a portfolio site and look for small web firms who need an entry-level designer. Some might hire you for boring HTML updates at first, but dues need paying. Most will not require an Arts degree, but they will want to see solid work samples -- sometimes that's all it takes.