Learning html and css, as mentioned, will be the basics you will use for building the framework of your websites.
To make them look nice you will want to work with Photoshop and Illustrator, and maybe Flash.
I like to use my desktop with a LARGE monitor for graphics. You might find (as I do) that a small screen is inconvenient when you have your workspace menus open and your design open on the screen. 24" screen or larger is nice to have.