First of all, I'm not exactly sure there are majors in Gaming Design specifically, at accredited Colleges and University. That being said, pursuing a masters and then a Ph.D. in Gaming design is most likely not even possible at that point.
Here's a good link at what I googled:
http://www.finegamedesign.com/education.html
You need to keep in mind that an education in digital art and programming is already established in many educational institutions but gaming design is not. Why? Well because gaming design is way too specific, all it does is "train" you specifically to do one thing in the computer world; build games.
If you are going to spend your money on schooling, then grad school, then even grad school again for your Ph.D. why not do it in something that can be used on a broader scale, say computer science.
Carnegie-Mellon has a good school for what your looking for I think:
http://www.etc.cmu.edu/
Cogswell:
http://www.cogswell.edu/computerandvideo.html
Then there's MIT:
http://orwant.com/mas490/ Last time I checked you need to be ridiculously smart and rich to get in there, but why not go for it.
The website I stated earlier is a good base but google it and do some research on your own. You got advisers at your school right? Use them! You got some math teachers and maybe some comp. people at your high school? try asking them a few questions. The resources are there its up to you to go out and look for 'em. good luck! and no, its not too early to start brainstorming the, arguably, most important years of your life.