Here is what I recommend Web Designers should know:
Dreamweaver
HTML
JavaScript
Photoshop
These are the core programs and will handle 95% of your needs.
You may want to add in Flash training, and if you decide to create web sites for other companies you should learn either PHP or ASP (web programming languages)
As in anything else, the more you know the better off you are. I don't have any specific recommendations for books, except the Photoshop Bible by Deke McClelland.
A great source for free information on HTML and JavaScript is http://www.internet.com/
