So for those who are interested in this path:
Games and movies are a good target industry for character design. You should know it's very competitive, and in reality you probably need to know how to do more than just character design - look into the very related fields of 3d modeling, visual development/concept design, and animation.
Colleges: In the US, there are many good schools with reputations for producing great illustration/animation grads: Art Center, RISD, Savanna School of Art and Design, and even San Jose State is known for it's art department. I'm sure there's more I missed. PS San Francisco's Academy of Art is a for-profit real-estate scheme masquerading as an art school.
These schools teach solid fundamental drawing skills, which you will need absolutely if you want to be good a character design. Don't get stuck drawing the same anime character over and over. You will need to be able to draw anything, realistically, and develop good observational drawing skills. Figure drawing is a must, and if you start practicing analytical figure drawing early, you'll have an edge over others who didn't.
If you look at the work of some of the great character designers in the West, you will see that they have excellent drawing skills that go beyond whatever style the character is drawn in. Check out Claire Wendling, Enrique Fernandez, the artists of Pixar, any "art of" books from Pixar, Dreamworks, or Disney.
Good luck to you.