Hi EYT,
It all depends what you want to create, as each program adds something different to the overall process.
Dreamweaver is the industry standard application for creating HTML based web sites. It will allow you to plan, layout and create individual pages, add interactivity in the form of javascript and ASP, and import many different file types. You can also manage and upload your sites from within dreamweaver. You're pretty much gonna need this if you want to get into web design.
If you want to create graphics to go in your sites you're also going to need Adobe Photoshop or something similar (i.e. a cheaper alternative like Paint Shop Pro). These let you create and export images off all types which you can use within your sites. Macromedia Fireworks and Photoshop 6 also support vector images (as opposed to bitmap) which can be useful if you want to create Flash websites, (which ill come to in a moment). Fireworks is also pretty good for creating animated GIFs (small animations you can use in HTML web pages). As far as I know, Ulead produce plug-ins for Photoshop, which are simply 'special effects' which you can apply to the images you create.
With an image program and a web page program you should be able to produce some pretty funky looking web sites, as long as you've got the talent! The only thing you aren't gonna be able to do is sites with huge amounts of animation, sound, games and more advanced interactivity.
For this you're going to need Macromedia Flash, I would recommend version 5 as its the latest. With this you can create colourful, vivid applications with loads of sound and animation that you can incorporate into your website in any way you choose. This allows for incredible diversity, with what you can do limited only by your ideas. If you want to create stunning, individual web-sites i would recommend learning to use Flash.
Coreldraw and xara are similar to Photoshop & Paintshop Pro but they mainly deal with vector images (although you can import bitmaps into them). Adobe Illustrator is pretty much regarded as the industry standard in this area. Vector packages are useful for things like magazine/leaflet layouts and logos because their 'vector' quality means they can be resized or zoomed to any degree and will still maintain smooth, crisp edges (as opposed to a bitmap which will become blurry/distored/pixellated if resized too severly). If you start using (the vector based)Flash a lot you may find a vector based drawing package like these useful as they offer more accurate and versatile drawing tools than flash, but can be directly imported into Flash.
The honest truth is that if you are serious about web design you're probably going to need one of each type of software to get anywhere good. Obviously choosing the big ones like Photoshop, Dreamweaver and Illustrator is going to be expensive but they are all excellent programs. However if you searc around you will find a whole range of cheaper alternative packages which do pretty much the same job..It just depends how much you wanna pay.
I hope that helps sort out your confusion a little bit! Good luck!
Nick (Web Designer)
nick.price@misuk.net