Unfortunately, at least in my experience, web DESIGNERS are about a dime a dozen. I am currently employed by a company and i do design as part of my job, but other skills such as SEO and programming have helped me grow within the company. Every once in a while i'll look online on sites like monster.com just to see what companies are looking for when hiring and employee in my field and most of them want a college degree (4 year, business or computer related) and about 5 years experience.
If i were you i'd start by building a portfolio online and posting on eLance. if you're just starting out and you don't really "need" the money right away, you should be able to find some design jobs you can bid pretty cheap on to help build up your portfolio and your resume.
On the other hand, the best designs in the world do little for a company if the website isn't found, and learning SEO is a great way to set you apart from other people who can only design.