I agree with most of what William has said but disagree about knowing flash as being a pre-requisite, it is just not generally accessible enough.
Also a good designer needs to understand content strategy (great books on content strategy are: Clout by Colleen Jones , Content Strategy for the web by Kristina Halverson and Mellisa Rach ).
I think the key thing to remember is that you need to start somewhere for me the basics are:
Server side scripting language (e.g. PHP) and associated database
Content strategy - with SEO as a subset of knowledge
Basic design principles, I absolutely love the 1st edition of the Principles of Beautiful Web design published by Sitepoint.
Principles of accessibility, a good place to start is on the Nielson Norman group website.