Most people are shocked to hear that I'm self taught, and some critics even ding me for it. The truth is that most computer experts are self taught, because this field is so young there hasn't been formal training until recently. I always tell my students that even though I am teaching them a certain version of a software tool, a year from now it will change and they'll have to learn another. Learning to teach yourself is one of the most highly prized skills a computer designer can have. Of all the possible backgrounds, I think the easiest to come from is multimedia, because it involves creating interactive content, working with 8-bit file formats, and awareness of platform differences.
Nothing is exactly like Web development, but multimedia development is the closest.
I think it's great to know a lot of things, but best to hone your strongest skills and interests. It's impossible to do it all well. I've never met anyone who knows everything, but some people know a lot more than others. I always encourage people to specialize in what they love, because its a lot easier to be good at what you like doing. I think it's great to know a lot of things, but best to hone your strongest skills and interests.