2 answers
Blaire’s Answer
Game design is one of those sort of amorphous fields that doesn't really have hard and fast requirements to get into. There are however a lot of skills and experiences that can help you become a better game designer. Going to college for Game Development is a great way to start learning these skills, but you will only go as far as you push yourself. A good designer is hired because of their portfolio, not their degree. That said, there are a few schools that are widely recognized for their Game Design programs. SCAD, Full Sail, DigiPen are some that come to mind that you should look in to. SCAD's been winning some student stuff at E3 the past couple years. A good portion of their program is artists, not designers, so while the design program itself might be a little weaker than some other places (and their programming instruction was almost non-existant when I went there) they do produce really pretty games, which look good on your resume even if potential employees are ostensibly looking only at mechanics.
These are some tips that I wrote out for another answer here, but I think they're relevant for you too. Like I said, you are only as good a designer as you push yourself to be. Classes will only take you so far.
1) Be skilled at communication, especially written communication. Anyone can be an "ideas guy", but it takes skill to take those ideas and break them down in such a way that your vision is communicated to the rest of the team. Designers are not islands, they must work closely with all the different disciplines (including other designers) that are coming together to create the game. One of the main things you will be doing as a designer is writing design documents, which depending on the scale of what you are designing for can be many many pages long.
2) Be a polymath (a person of wide-ranging knowledge or learning). Games as media range over a huge number of subjects. Because games are reflections of real world experiences, a game designer must have some knowledge of those experiences in order to make a compelling game. To a game designer, there is very little knowledge that could not be potentially relevant. History, biology, engineering, physics, art, music theory, psychology (especially psychology); you don't have to be an expert in every field, but working knowledge of a wide variety of topics can help you at least identify what you will need to explore in more depth. As a corollary to this, always do your research. This doesn't mean that you necessarily have to go to war in order to make a war game, but interviewing soldiers who have had that real world experience will help you find ways to bring depth to your game that you wouldn't have had otherwise. Knowledge is the birthplace of inspiration.
3) Play games like a designer. When playing games, examine the design decisions they made and how effective they are. If something is boring or frustrating about a game, try to identify what weaknesses in the design are contributing to your negative opinion. If you're spending hours playing it, step back and examine what about the game is drawing you in. Is it truly engaging, or is it stringing you along endlessly with reward loops? Play games not just for fun, but for studying. Play games you wouldn't normally play, genre's you aren't drawn to, kids games, "girly" games, casual games, browser games, tabletop and board games, right along with mainstream AAA. Too many new designers limit themselves to the games they like to play and can miss out on understanding huge portions of the market. After all, those market niches are there for a reason. I'd recommend watching this Extra Credits episode (part one and two) as they break down why this is important and how to do it much better than I can in a single paragraph. Actually I'd recommend watching that youtube channel in it's entirety. It's great thought material for aspiring and experienced game designers alike.
4) Take game design courses. It's true that you can be a very talented designer without ever having set foot in a game design program, but most reputable design courses will bring you through a lot of important topics that aren't really covered elsewhere or are only really relevant in the context of game design. Things like how to employ random elements without making the game luck based, design methodologies that simply wouldn't come up in any other area of study, research done about games and gamers, managing project scope, relevant psychological concepts like reward loops, flow, and creating illusions of choice.
5) Script. You don't need to be able to script in order to be a designer, but understanding code logic and how to make events happen within a game engine will make you much more valuable to potential employers. I'd really recommend doing some unity tutorials to get a handle on C# or use UDK to play with visual scripting and blueprint, which is just C++. It will also allow you to make more games in a digital format and experiment with ideas without having to ask for outside help.
6) Make games and make lots of them. This is the most important thing you can do. Practice really does make perfect, and the more examples of good design you have on your portfolio the more you can convince people you know what you're doing. Your games don't have to be pretty, but they do have to be fun. A well designed game that uses only gray cubes is more valuable on your portfolio than a pretty game with awful design. When you're starting off, don't make the mistake of planning big open world RPG's or complex multi-player experiences. Your first few games should be small and simple, but polished. Branch out as well. Board games might seem like they're not relevant, but the skills are transferable and a published board game is a great portfolio piece.
Brock’s Answer
There are many good colleges out there for video game design. I can recommend my school Full Sail University. It is one of the top 10 design schools, according to the Princeton review. What you major in is really up to your preference. There are so many different things you can get into from UI/UX to a technical designer.