Some of the better commercial 2D animation programs such as Digicel Flipbook, Toon Boom, TVPaint, TAB, and PAP, have Student or Express versions that are fairly inexpensive. However, you might be disappointed by how crippled these stripped down programs are compared to their more expensive Pro versions. But if you look hard enough you should be able to find them cracked online (rapidshare, bittorrent). There are some free alternatives.

If you're doing old school drawn on paper frame by frame stuff or stop-motion or 2D, I'd suggest MonkeyJam. It's free.

http://www.giantscreamingrobotmonkeys.com/monkeyjam

If you're on a Mac, use Pencil(there's PC and Linux versions as well).

http://www.les-stooges.org/pascal/pencil/index.php

If you want to do Flash animation like the stuff on Newgrounds.com, either break down and buy Adobe Flash or you can try Creatoon or Synfig, which are free. They're not Flash, but they also do vector based symbol oriented
animation.

http://www.creatoon.com/index.php

http://www.synfig.org/

Here are some online resources about the basic principles of animation.

http://www.idleworm.com/how/index.shtml
http://www.awn.com/tooninstitute/lessonplan/lesson.htm
http://www.animationarchive.org/2006/05/media-preston-blairs-animation-first.html
http://www.animationarchive.org/2006/05/media-preston-blairs-animation-1st.html
http://www.karmatoons.com/drawing/1a.htm
http://www.animationarena.com/principles-of-animation.html
http://www.animationmeat.com/notes/notes.html