I’d have to agree with David, it is tough as it used to be just build a site and that was it (pre 2000) then the odd site update here and there.
Now you have to have a fair bit of business know how, as some companies will want to hold you accountable for what your doing and see a return on their investment with regards to the website.
Also a lot of businesses will be aware of their standings on the search engines. One wrong move by a web designer and you can be responsible for their site slipping down the rankings or even worse dissapearing completely.
I really don’t want to scare you, but it’s best to know now rather than be sitting in front of a couple of MD’s trying to explain why their site is responsible for a 10% drop in sales, and for a lot of small to medium sized companies these days that 10% is usaually a 6 figure sum.
If you’re just starting out you really need to know your basics, xhtml, css and making sure everything is as standards compliant as you can get it. Then do some reading up on internet marketing / seo.
Once you have that nailed you can then decide which way you want to go with regards to the nice to haves (PHP, Flash, more internet marketing, mobile design, or just web design for the front end stuff etc etc)
As for working self employed, I do exactly that on top of my day job, it’s fantastic when things are going right but you also have to be responsible for any mistakes you make and take care of general day to day business (phone calls, accounts, marketing, meetings etc). It’s very rewarding but can be stressful at times.