Whilst having a natural flare for web design will help you up a few steps, it's possible to learn it. One step at a time, though, I would recommend. Learn HTML and CSS, then learn a bit of javascript, then download mootools and save yourself about 10 years every year. Don't worry about your talent as a designer, because for most intents and purposes, designers know nothing about designing for the web.
Your knowledge in photoshop will probably help you. Knock up some simple wireframes (just coloured boxes, positioned all weirdly) then make that into HTML and CSS to teach yourself what is possible in the various browsers you'll be supporting. Once you're able to confidently, quickly and accurately convert a template to an HTML and CSS page/set of pages, start to transpose your way of working to designing in-browser. Because the market is moving so quickly at the moment, and the browsers it's still relatively important to support continue to be very popular (but archaic), I think it's important to present your designs in the manner in which they will end up. A PSD template in some respects is a pipe dream, and if you find that something can't universally be achieved, then expectations aren't met and people get disappointed.
Buuuuut, I'm getting way ahead of myself.
As a big aside, that saying "jack of all trades, master of none" is very true. Your life will be a lot easier if you learn simple scripting techniques like server side includes, but application development and interface design are akin to car concept drawing and car manufacture. A designer will have to have a handle on what is possible, but they won't need to know how to do it - it's not their job! That's not to say that you can't be a good designer and a good coder, but you probably want to be great and I'm yet to meet anyone who is really great at both. Like I said, it's not impossible, but they're two totally different kettles of fish.
The only really grey area here, to be honest, is the whole "do developers do HTML and CSS, or is that a designer job?" - the answer, in my experience, is "both should be able to do both to a high standard". If you call yourself a web designer, but you don't know HTML and CSS, you basically draw pictures of websites. Some people make a career from that, but I bet most of them aren't very busy!! If you're a web developer and you don't know HTML and CSS, that code you're writing, what exactly is it doing?!
I've waffled on too long, you basically only need the first 2 paragraphs, but I've written too much to delete all this now!
