Well... What do you want to publish to?
If you plan on publishing to web, Flash is something you might consider. Although, I must mention, it is not designed for cartoon animation per se, it is more for dynamic web content and web content in general. If I get the right feeling from your post, Flash is not a good reccomendation.
If you plan on publishing to film or video (or even to web, ie ifilm) some popular ones include ToonBoomStudio and Mirage. There are a number of others, just google for 2d animation programs or something to that effect.
http://www.toonboom.com/
http://www.bauhaussoftware.com/
If you want a good product, you're going to have to spend a decent amount of money. Often times if something is free, you get what you pay for.
As for getting started... get some books. Look for tips on the net, web forums, etc. I would also look into the history of animation. Learning how the original masters learned can be extremely beneficial.
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