Alright, as far as colleges/universities go I hope you're in the USA otherwise this won't make sense. Let me introduce you to the research and findings of my past 2 years.
So, Game Designer? Is there a specific part of a game that you want to get to learn? I mean you can argue how it's better to specialize in one thing or be a jack of all trades going into the game industry but you're going to work as one thing at a time. Digipen is good, they've got great alumni for proof (Portal came from there).
I'm going to discourage getting into the Art Institutes as I've heard and have friends who have been actively ripped off. I can't tell you that about each AI though (they all vary). I can specifically tell you more from an artist's angle (since I'm aspiring to be a game artist). Look into SCAD (Savannah College of Art and Design), Ringling College of the Art and Design, and RISD (Rhode Island School of Design). If you asked for my opinion I'd say look at SCAD or Ringling. They're the two big names (apart from Digipen) when it comes to Game Art. They're even Art Colleges themselves so you can explore other opportunities there. I've applied and will attend [at some point] SCAD. I really like their program in which although it's an art college, they give you experience in programming and designing/managing the games as well. This way I'm not locked into being just an artist (though a degree in Computer Science would serve better than learning programming from an art school).
So other side of the spectrum - the programming. As far as the programming side goes, you can find a college specifically tailored to turn you into a programmer for games or pursue a degree in Computer Science to become a programmer overall. Deciding to get a degree in Computer Science may lead out to be for the better since you won't be locked into just being a game designer. I'm thinking of Job prospects here. I'll throw out a big shot Ivy League school here and point out that Penn State recently made a separate Game Design program. Oh this leads me to a common misconception..
Game Design being a relatively new field to actually go get a degree in, all schools call the major itself different. The only way to tell what you're learning is really digging up the website and alumni, looking at student work and reading reviews and finding out what you're learning. Penn State's Game "Design" program is basically a branch off of Computer Science.
If you're looking for study abroad opportunities I'm just going to slap down the name NHTV right here. It's a super super super good media university in The Netherlands. A really good friend of mine is attending there. The way they have their system broken down is that you pick the specialty you want to go into (Visual Arts, Programming, Management, or Indie Development) and they vigorously break your back until you know your pipeline inside out, have had internships and have gotten job offers while you're still in school.
And if you're like me - wanting to be a game artist and too impatient for college (or want a head start) consider self-studying. You're going to be Self-Studying even past going to college. This is a field that's constantly changing and developing and so you need to keep up. I'd recommend a site such as digitaltutors.com. Buy yourself a month or so and fiddle around to see what it looks like. I got in there and really took advantage of my summer last year with that.
Furthermore, let's say you're not sure about what college to go to and don't know what part of the actual game creation you like. I'd recommend looking into the topic even more. Look at the pipeline, the game development, the actual stuffs that go into the creation.
Sorry I can't tell you more. "Game Designer" is a vague term really. What do you actually want to do?