With programs like Macromedia Fireworks MX, or Photoshop it is pretty easy to make animated GIFs (which is what you wanna make for an avatar)... I'm sure there are cheaper apps available too.
I make mine a little more complicated by using morphing software, Adobe Premier (for effects and transitions), and Fireworks to clean 'em up. But, this path can get a little costly and takes some gettin' used to.
Start out with an application that can create frames and export animated GIFs. You will find this is a bit tedious for large animations (many frames), but cheaper and easier than other methods.