I got started with the old Sun tutorial which had a page on animation. It has been retired but (they say it) is available for download on this page:
Download the Java Tutorial. I think you want the Online Tutorial link. The rest I learned on my own.
For background information on painting you might check the currrent tutorial page:
Implementing a Custom Component.
Here's an example of animation in Swing:
[ June 27, 2004: Message edited by: Craig Wood ]