There are lots of ways to get the education and experience needed to become a professional Web designer or developer. But there are some basics that you should know in order to get a job so that you can gain the experience needed for more advanced jobs.
Basic Web Development Knowledge You'll Need
Some people will tell you that because WYSIWYG programs are so wide-spread, you don't need to learn HTML, but unless you're going to stay in business for yourself, eventually you will come across a hiring manager or firm who wants you to prove you know HTML. Beyond that, HTML is the backbone of Web design, and if you know how Web pages are put together, you will be better at the job - even with a WYSIWYG editor.
Cascading style sheets are what make your pages look good. And even if you're planning on doing more Web programming than Web design, you should know how CSS works. The content and behaviors of the Web page interact with the CSS to create the full design, and CSS can become very complicated.
Keep in mind that when it comes to general education and experience, most large companies will want you to have a Bachelor's degree. Small companies don't care as much, but they also don't always pay as well.
But that's not all you should learn. Web development jobs often require or request that you have other education and experience, depending upon the type of job you're applying for.
Web Designer Education and Experience
Web designers should focus their education on design - graphics and layout. Most companies hiring designers want people who are visually artistic. You should study color theory and composition and get a degree in visual arts or visual design.
Focus your education on design and less on building Web pages specifically. The sad fact is that most Web designers have spent a lot more time learning HTML and how to use Dreamweaver than they have learning anything about white space and creating a design that flows. If you get educated in classical design techniques and skills and then learn how to apply them to Web pages you will stand out as a designer.
Most companies looking for Web designers will want to see a portfolio of sites that you've designed. Be sure to keep screen shots and color prints of the designs you've worked on - even if they were just class projects or sites you built for yourself. Try to have a diverse portfolio that shows more than just the front page of any site, and remember that your designs won't remain on a site forever, so keep your own copies.
Web Programmer Education and Experience
Web programmers focus on the behavior of Web sites - many companies don't hire Web programmers specifically, but rather software developers who are skilled at a specific programming language. The most common languages used by corporations on the Web are: PHP, JSP, and ASP.
Web programmers do best when they get a computer science degree. It used to be possible to get a Web programming position without a degree in computer science, but the level of programming required for most enterprise Web sites demands highly skilled computer science professionals.
Don't focus on any one programming language. Chances are, by the time you finish school, that language will be "out" and something completely different will be "in". Companies follow fads just as much as any other industry, and Web programmers need to be aware of what's hot and not. You're better off learning how to learn programming languages and then scanning the jobs 6 months or so before you are going to start working to find out what language you should focus on to get hired. Some good bets right now are: ASP, JSP, and Ruby. PHP is popular with smaller companies, but has a lot of security issues.