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If you're looking to get into 2D computer animation, try TOONBOOM STUDIO or BAUHAUS MIRAGE or THE TAB. Mirage is really good, although really kinda dense when you first start out, and it basically operates as digital paper, allowing you to essentially use Photoshop on every frame to rough out, sketch, ink, paint, etc. each one. THE TAB and ToonBoom, on the other hand, operate more like Flash, allowing you to create, reuse, and reshape drawings each frame.... In a comparison of Tab and ToonBoom, I prefer ToonBoom, I like the interface more. Mirage is great if you give it a chance, though.
Best part about these three? If you follow the links I gave you, you can download trials of each of 'em. Sweet, huh? :)
If you're looking to get into 3D, try HASH: ANIMATION MASTER, it's really cheap, but really really good, it even has hair, fur, stuff like that. Only downside is that it models using 'Hash Patches,' which aren't really used anywhere else, and it can be a little weird switching over to Polygons or NURBs, but as long as you don't develop any bad habits modelling in Hash, you're good to go as soon as you pick up another program.... I would totally reccommend this one, though, if you're looking to start some 3D animation. At a mere $299.99 price tag, how can you afford not to? :P