A Computer Science major would prepare you to be a game programmer, but it isn't necessary for a game designer. Game design usually does not involve any programming whatsoever. It's done entirely with writing and math, and then communicated to the programmers, who implement the designer's ideas in a programming language like C++. It's the designer's job to come up with the ideas for every tiny detail of the game: what NPC the player encounters where, and what the NPC will say; what weapon the player can get at what point in the game and how much damage it does; what attacks a boss mob has and how much damage they do, etc.
If you're interested in programming for games, Computer Science is the way to go. For design, you can major in anything you want, at any good 4 year college. I've worked with game designers with degrees in everything from Theatre to Biology, but liberal arts majors are probably the most popular. You can major in Computer Science if you want, but it has it's own pros and cons: game programmers can often get hired right out of college, whereas designers usually have to start out in Quality Assurance or Customer Service and work their way up to the design team. OTOH, being a programmer can sometimes pigeonhole you, and make it difficult to move to the design team.
Whatever you decide to major in, also take math up through Calculus 1, and at least two courses in Statistics, two courses in writing, one Computer Science course (if that isn't your major), and one art course. Math and writing are the main tools of a designer, and the CS and art courses will help you work with your programmer and artist colleagues later on. Fill up the rest of your course requirement with a wide liberal arts education -- history, literature, mythology, sociology, psychology, etc, are all useful in game design. Generally speaking, game designers need to have a wide education, rather than a narrow Computer Science education, so if you do decide on a CS major, be sure to take lots of writing classes, and as many liberal arts classes as you can.
If you want to get into any form of game development, if at all possible go to college near city with a lot of game studios, like Los Angeles, San Francisco, Austin, Seattle, etc. Most game companies will not pay for relocation for an entry level job, and some won't even interview you if you don't live in the area, so it'll be a lot easier to find a job after college if you already live near several studios. There's a map here: http://gamedevmap.com/
that lists every game studio by city. Try to find a college in a city with at least 10 game companies.