Even though I went to the Guildhall at SMU for level design, Full Sail is another excellent school (though I know little about their Game Design program). As with any school, always make sure to do your research on the school. Visit the school and make sure they answer all the important questions/
Who teaches the classes? It's pretty essential that someone who is teaching you actually has experience and a lot of it. A lot of schools will hire people straight out of school (I got approached by a few schools immediately after graduating) or get people with 2-3 years experience. Make sure the people who are teaching you were at the very least senior level and preferably lead level.
What is the placement rate and, more specifically, what is the placement rate in the students chosen field of study? A lot of schools will tout a high placement rate but they're counting people who went their for design and wound up as a tester. Testers are absolutely essential and important but you're not going into debt to become. Make sure the placement rate for the school is really high, The Guildhall was something like 94% when I was there.
Make sure the program focuses on design. As a designer, it's good to have a wide spread of eduction but make sure the program focuses on the design. If you're going to school to be a designer don't have delusions that you'll double as a programmer or artists. It's good to have basic programming skills and it's great if you can make proxy models, but you generally won't be actually programming and you won't be making final art.
Try and talk to their placement adviser and see what they can do for you. With luck, they will say they will help you make contacts but it's up to you to land the job, that's the right answer. Run away from any school that guarantee's placement or says they'll do all the work for you.
Don't go to a school that has a commercial. This isn't a question but it's important. This is aimed specifically at places like Devry and Westwook college. If you paid attention to the above questions, you should be safe but just be on the lookout for schools that only want you money.
One final note-- being a designer is all about your experiences and the more experiences you have, the better designer you will be. I don't know if Full Sail requires a degree in anything before hand, but it's good to go to college and then go to a place like Full Sail. Get a degree in something you're interested in and take a bunch of history electives. Or art electives. Or programming classes. They're all incredibly useful and can only make you a better designer. You don't need to rush into the games industry, it will still be here in six years.