If you want to be a game programmer of any kind, getting a CS degree is the right way to go about that. Game-specific colleges are no where near as well respected as "real" CS courses.
If you don't want to be a programmer though, then CS is not the course for you.
If you want to be a designer, I'd not advise setting that as your first career goal. Studios often have one full time designer per two or three dozen other staff. It's an extremely rare position, and it also requires a lot of experience in other areas, so it's an extremely competitive rare position. IMHO typically a designer should have in-depth knowledge of one hands-on game development skill (e.g. has been a programmer, environment artist, etc, etc) and experience as a producer or project manager. That's not a job that you hire graduates/juniors into. Being a game designer is also a hell of a lot harder and more complicated than most people imagine. It definitely is not a job about just coming up with ideas, but often much closer to a project manager's role.
In any case, getting an education alone is not enough to get you a job in the industry. As a game programmer, you should already be programming games before you start your CS degree (so you know whether you enjoy it or not), and continuing to do so in your spare time for the next 4 years. If you want to be an environment artist, you should be practicing making 3D environments for the next 4 years. Or if you want to be a designer, you should be designing games for the next 4 years -- note that even without programming/art, you can create a card/board game to completion (and should do so). If you want to be a designer, it's worth learning some basic code/art skills so that you can pick up something like Unity and make prototypes of your designs by yourself too.