The career path for a 3d animator or game designer is extremely long and difficult. You will have to start out at the very bottom of the totem pole so to speak. Making simple 3d images and digital renders of basic objects, with simple textures is the starting point for most digital artists. This type of career path takes a very long time to travel, and it also requires hard work and patience. It's not something you can go off half cocked and succeed at; you need to work hard at this. The starting point is usually a trade or tech college as most community colleges do not offer courses in these fields.
Recently design jobs in the industry have become a highly sought after position as many game companies are starting to make new games every 2 years or so. Designer jobs for a 3d studio or art position are also highly sought after. Although since many people want these positions there is currently an over saturation of artist's in the market. In order to stand out and get hired you need something that makes you stand out from all the others. A simple two year degree will not be enough.
Most searches for the words jobs graphic will show you links to school were you can get a degree in the field. This can be helpful if you are currently searching for a school to attend. It's important to keep your skills up to date in this industry as newer versions of programs are constantly released, and methods of doing things are updated. Most 3d modelers can attest to this. This is where patience plays a big role in this career path.
Many aspiring 3d animators and artists fail because they think they will be taught everything they need to know during school. This is not the truth in most cases, if you want to truly be masterful at something then you need to work at it, and learn independently after acquiring your degree. If a company sees you are willing to do this continually chances are you will be hired before everyone else.
Again it's important to have some kind of edge over everyone else. Having a good degree is one thing, but in the end it's a piece of paper. If you can't back up what you say you can do then you won't get hired anywhere. You have to work hard and make something truly amazing, after all companies want the best. So wouldn't you if you were in their position.
As you go out into the world on your career path searching for design jobs, or designer jobs keep in mind there are many others out there competing with you for the jobs? It's a highly saturated market so work hard and you will find your position. One thing to consider is to never be static, always keep learning, and stay up to date on everything related to your field. Make yourself as indispensable as possible.