Probably one of the more important things to have a grasp of for a "graphical web designer", beyond your illustration tools, is CSS (cascading style sheets). It saves a lot of repetition, and in modern browsers you can get some pretty impressive effects by combining some of the more recent CSS attributes to different elements. If you'll potentially be doing more than filling a graphic designer role, you will probably need to be familiar with basic (X)HTML and "tableless layouts", some knowledge of SQL, and at least one scripting language such as PHP, ASP, Cold Fusion, or something similar.
Above all else, the most important trait to possess is flexibility. Between client whims and browser quirks, you'll go crazy if you're not able to adapt to changing situations and requirements. In some ways, (correctly) designing for the web is far more of a challenge than writing game code.