Writers for games is still a pretty select group. A lot of games don’t have writers at all, and many many others hire only on contract for a short time, rather than keeping a writer around full time.
That said, there are two basic routes to get to your goal of writing for games. The first is to become a writer in another industry, like comics, movies, books, etc. Game companies almost never hire a writer without published titles to his or her name. For this you would need the typical writer education – Creative Writing, English, or Film Studies, usually.
The second route is actually to become a game designer and hone your writing skills to the point that you can act as a writer as well. A lot of game companies prefer this, because for six months they can have you working on story and characters and dialogue, and the next six months they can have you working on naming items in the game, for instance. It means a full-time salary for you, rather than contract work, and it means that they don’t have to go looking for a writer.
Pure game design, aside from writing the plot and dialogue, actually involves a lot of writing itself. Game design is the process of coming up with all the small details of the gameplay, refining the math behind every mechanic to make it fun and balanced, and then writing the whole thing up to communicate it to the programmers, artists, level builders, producers, etc. Game designers also typically name items (and NPCs, if there isn’t a writer) so knowledge of words and literature is very helpful as well.
If you want to go this route, I would still recommend either a Creative Writing or pure English degree. You should supplement that with math as well, by taking math classes up through Calculus 1, and at least two semesters of Statistics. Take one art class and one computer programming class, so that you can work better with your artist and programmer colleagues. Beyond that, 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.
If at all possible, go to school near a 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. If there’s a specific company you want to work for, try to go to school near that company’s headquarters.
One last thing to keep in mind, if you go the designer route: almost no designers, and no writer I’ve ever met, have been hired into the game industry as a designer or writer as their first job. Most designers start in another department, usually Quality Assurance or Customer Service, and then work their way over to the design team. And like I mentioned above, most writers hired by game companies already have a published title in another field. But if you work hard, perfect your art, and do what needs to be done, you can absolutely get a job writing for games.