First, IT support and Web design are two different fields, and although there is some overlap between the two, you will have an easier time if you clarify what you are going for before you start.
Web Design is distinct from Web Programming and Web authoring. Web design focuses on graphic design, look & feel, layouts and fonts and things. If you are a highly aesthetically sensitive person, Web design is a good field for you. Warning: you might become irritated by bad kerning on road signs.
Web programming is programming. It’s like any other kind of programming, just more popular. There’s a lot of overlap between what you learn in setting up your programming environment (httpd, database engine, CGI or other runtime setup, VCS, editors, the whole deal) and the necessary knowledge for certain types of IT support. My first support job was for a VPS server provider, so my customers were largely website admins who had questions about apache, MySQL, PHP, and all that sort of stuff. Later I worked for a large online retailer, for a computer store, and for some offices- none of these made use of my web-programming and sysadmin knowledge. Mostly I needed to know about Active Directory, adding printers and such under Windows and Mac, and other office-type things. It can be hard to predict in advance what kind of knowledge will come in handy for what kind of job, but it’s also sometimes hard to learn it ‘on the fly’.
Don’t learn specific software, learn languages and protocols. In a few years Indesign will be as dead as Macromedia. Learn HTML5 now, and you’ll still be in good shape when HTML6 comes out.
Here are some things you will need to know:
The OSI layer model of networking. Memorize it. Do it now. Expect to be quizzed on the OSI layers at any tech-related job interview.
Learn HTML and CSS: Learn to code it by hand. Follow standards, don’t depend on special browser behavior. Read alistapart.com.
Learn at least one programming language. Also learn Javascript. I don’t recommend you start with Javascript, as it encourages bad habits, but what the heck: I started with LOGO.
Learn XML. It’s hard, but you’ll be glad you did.
If you are a Windows user, learn Powershell. If you are a Mac / UNIX / Linux user, learn bash. Actually, even if you’re a Windows user, install Cygwin and use bash.
Learn how relational databases work.