Dear Career Games:
We do not know what your current background, skills and experience are, as they relate to your career objectives. In any case, you are not likely to find colleges that give degrees that relate directly to video games. You will however find colleges that offer the foundation of comprehensive knowledge on which you can build the skills necessary - through on-the-job experience or self-guided learning - to design and build specific applications, of which video games is one. These programs are computer science and computer engineering, which many colleges offer.
As you may know, video games come in many flavors. Hardware platform based video games such as Microsoft X-Box or Sony Playstation have been designed by dozens of hardware and software engineers. Third party applications designers employ software engineers to develop more games for these platforms. Same is the case for stand alone games, such on the PC platform.
If you are interested in designing the platform for a video game, you would need skills in the hardware design (and software design if you wish to design its operating system). You would require software skills if you wish to develop video games as applications on a given platform. It is believed by many people that many video game designers do not have formal degrees, that they are either self-taught or have obtained on-the-job experience. This is true and it is not true. Some of the most successful game designers have degrees from some of the best universities, sometimes in non-computer-related fields. And some of them never went to college.
Since you are seeking formal training, a program in computer science or computer engineering will stand you in good stead. Then you can pursue the video games arena, or other choices that would be available to you.
As a final note, you should go to the website for Tom Sloper, who has much exerience in game design and who now teaches and advises people like you who are considering a career in game design.