Materials

There are a million defferent mediums to use for drawing, but here's a simple process to start with:

1) sketch in pencil, 2) go over it with a pigment liner and erase pencil lines, 3) color with colored pencils, markers, or paints.

• Pencils - Anything probably works equally for sketching. Mechanical pencils are cool. School pencils are fine, too.
• Pigment ink - I have a couple pens that are called pigment liners, and they're basically the same as small sharpies. The key is they don't bleed when they get wet. Ballpoint pens smudge, so you shouldn't use those. Very useful for outlines!
• Coloring - Go crazy! Use whatever you'd like. Sometimes something I like to use are watercolor pencils: they're just like colored pencils, but when you go over them with a wet paintbrush they turn into watercolors. They make the coloring smoother and nicer. Good for people like me who don't have the patience (or skill) to do good smooth shading with regular colored pencils. :)

Programs

For any digital drawing (as well as graphics for my site), I use Jasc Paint Shop Pro 8. Now that Corel's bought the company, though, who knows what's happened to the program... =P You can probably find PSP 8 and 9 on Amazon or eBay.

A more common (and more expensive) program is Adobe Photoshop.

For free there's GIMP, but I've actually never tried this program before. ^^;

More help

If you really like drawing and are not content with my puny little How To Draw page, then you should get a book. Seriously. :D I happen to own this book, I got it in like 5th grade: How To Draw Manga, by Katy Coope. It's for beginners and gives you a good foundation in drawing anime-ish stuff, and from there you can develop your own style.

Be observant!

Something that will really help you in drawing Avatar is to keep your eye on the animation as you watch. =) Look out for the little things! For example, notice how Aang's pupils get really tiny when he's surprised, the shape of Zuko's right ear, how Sokka's ponytail looks at different angles, etc... If you understand the details of each character design it'll be much easier to draw yourself.