I do much of my graphics work in Macromedia Fireworks MX for web applications and Illustrator (if I’m working on my Mac) for print applications, and any photo editing in Photoshop CS2. Here’s a few of the sites I’ve designed, with almost all of the graphics work done in Fireworks MX:
http://www.team228.org/ (I designed the everything (graphics and PHP/MySQL database-driven backend) from scratch.)
http://www.platths.com/ (I designed only the graphics, not the ASP backend.)
http://www.connvex.org/ (I designed the everything from scratch.)
I avoid Flash [as part of the main design/navigation/framework] of any website like the plague. Not only will it cause a lot of problems with things like search engines and text readers for blind people, but it usually wastes more bandwidth than it’s worth. Good uses of Flash might be embedded Flash video players, or an interactive presentation about your team on a dedicated page. Bad uses of Flash include links/navigation and “useless” animation fluff.
(Useless animation just distracts people from the content of your website, which is far more important than bright and flashy websites.)