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Being an independent webmaster, I have a lot of different hats to wear, finding myself being a jack-of-all-trades and master of none. I have yet to seriously delve into dynamic sites that use databases and scripting. But from a static-page standpoint, which is probably where you want to start, focusing on HTML, graphics creation, and site colors/layout/usability should be first on the list.
Some well-known apps, sometimes considered industry prerequisites, are photoshop and fireworks for graphics and dreamweaver for html. I agree with David's comments on getting a firm grasp on HTML before using WYSIWYG editors like dreamweaver. I use homesite too, and it is great for learning html. (But notepad can suffice if necessary.)
More advanced areas, often sought out by employers, are ASP, PHP/MySQL, and PERL. These are usually needed for dynamic sites and companies that showcase/sell many products.
Flash is also becoming very popular for rich media type presentation.
With all of the different areas you can get into, it may be wise to take them on one at a time, and maybe pick from these to develop specialty skills. You may want to focus on technologies that are more intuitive to you, or that you enjoy working with.
A laundry list of apps and abilities for a well-rounded, experienced webmaster might include some of the following:
website marketing & promotion
OS proficiency (winders, 'nix)
Please note, I doubt anyone is proficient in all of these. This is just a list of some things that are good to have in your arsenal, and look great on a resume. No one should expect a webmaster to know them all, so don't be overwhelmed by this list.
I am sure I have omitted some of the obvious. Hopefully someone will come along and fill in the blanks.
Basically, I think a webmaster needs to be able to build, maintain and promote quality websites. And considering how fast the web changes, you can never stop learning how to become a better webmaster.
I personally started out designing small sites for myself and family/friends using GoLive (an alternative to Dreamweaver). GoLive allows you full access to your html code (like Dreamweaver), and over time as I tried different techniques for page design, I spent a great deal of time tinkering with my code, and learning HTML. In retrospect, I would recommend building some very basic site layouts using straight HTML before using a WYSIWYG editor though... it really would have shortened my learning curve if I had.
Specific software requirements don't really exist. Pretty much everyone will agree on PhotoShop for grpahics... but some people also swear by Fireowrks, which I don't use. Some people say Flash is the wave of the future, some people use LiveMotion to create Flash files, and some people don't do Flash at all. For code, some purists insist plain text editors like notepad (Windows) or BBEdit (Macintosh) are the ONLY way to go, some people use WYSIWYG programs like Dreamweaver or GoLive, and some people use other programs I know nothing about. I use a combination of GoLive and BBEdit now, depending on what I'm tyring to do at the moment.
If you're trying to get a job with a specific web design company, find out which programs they use, and learn them. If you're trying to work freelance and start your own business, use what works best for you.
Basic skills (IMO) would include:
Designing quality graphics and being able to optimize them for internet delivery,
Knowing the basics of getting a site listed in the search engines and directories
I learned as I went along. I didn't start calling myself a "professional" until I was fairly confident I could do a better job than a lot of the "professional" sites I saw online.