Good web design begins much earlier than Photoshop/Fireworks, and doesn't even move to the graphics/proof stage until you've made dozens of decisions beforehand that will influence your design (think business goals, marketing messaging, platform, audience etc.) HTML/CSS is almost one of the last steps in a complete web project.
I would definitely create buttons or icons in Illustrator (but you still don't have to do that), and do a lot of everything else in Photoshop. You can try Fireworks and see if it's your cup of tea, but Photoshop definitely isn't going anywhere as far as standard practice for awhile, at least from what I've seen. I've worked with startups all over the country and top web development firms and still all I see is Photoshop/Illustrator.
Not to say Fireworks is bad or you shouldn't use it, but take it as a grain of salt and see what works best for you. It never hurts to learn new things.
Ctrl/Cmd+click with Photoshop move tool automatically selects the layer that is under your mouse cursor raster. Not 100% Illustrator-style but less BS than selecting from lists.
One other thing I do is try to make sure I complete my web design without having to use any image or Javascript. When you limit yourself in this way, you find ways to accomplish in CSS/CSS3 and get better at it eventually.
I've only been at it about 5-6 months, with 1-2 years experience in graphic design. Any advice?