I just got hired as a game programmer although I never directly planned to get into this particular field of Software Engineering.
You should study:
Software Engineering Major (or equivalent)
Various Programming Languages: C, C++, C#, Python, SQL (and as many more as you can manage to learn)
Database programming (SQL)
Logic and Design
Mathematics: as high as you are capable of going
** Study as much on your own as you can-- this is very important-- school will never teach you enough and languages and technology are ALWAYS evolving and so should you.
Learn, have fun, and create some cool games! You can do it and make nice money\ benefits while doing so.
The biggest problem I find Scott, is graphic artists. I can write code, that isn't the hard part..... the hard part is getting people who can draw on the computer and/or import their pictures to an electronic file.... remember, sprites are a lot like a flip-book, and even rendering requires graphics in order to work. I've built games, but without a team of graphic artists, it's only a piece of code with moving pictures....
I completely agree with Mr. Comatose without graphic artists and animators there is no point in moving ahead since there is nothing you can test your code on. Map developers and Graphic artists are the key to game dev. Round up some of your friends who are interested in graphics if you are going for indie game dev or look forward for joining a company which is much safer bet.