I operate a animation and design business and 3D stuff takes up the main amount of work.
I personally use LightWave, amazingly good program for the price, yet would not be the best program to start off with if you are just starting out.
The first program I used, and I would recommend to get the basics of animation is Swift 3D, this program is rather basic yet introduces you to a lot of concepts, Swift 3D is designed mainly for vector and webpage work, not for film, tv.
Another program to try which apparently can give the end users good results for little work is a program called Carrara, and has been said to be a good learning program, and for the price is extremely good aswell.
Maya is the industry standard for high-end graphics, would not recommend giving this a go if this is your first time.
3ds-max is the industry standard for game modelling, and is another program I used before I switched to my current program LightWave, 3ds Max is an amazing program, and is also used an awful lot for film and TV. 3ds Max is by far the most used and most pirated program in animation.
LightWave is still used a hella lot for TV, did the graphics for Firefly, Battlestar Galactia (?sp) and also for a lot of movies, was used an awful lot for 300.
Just remember most movie studios do not use one package alone, they can use quite a few, using each of them to their advantage. LightWaves modeller and XSI both are apparently extremely good modellers for example.
Some studios (Pixar for example) write their own software aswell which isnt available to the public, and have teams of coders working on new technologies and developments all the time.
If you want to stay legal but dont want to spend money, Blender is the way to go, free open source program that has just released its first Open Movie, Big Buck Bunny, visit Blender.org for more info on that.
PM me if you have any questions specifically about animation or LightWave and dont want to ask the forum.