I would recommend studying general computer science, even if your desire is to become a game developer. You do not necessarily have to take the 'hardcore classes' of computer science, but basic knowledge of how programming, computers, etc. works is important. Especially for a game developer, who needs to know how much (or how little) he/she can ask of a computer.
A fair grip on user interfaces is also important. While a lot can be learnt by experience in this field, getting the right mind about how your users interact with your program is very important. Bad user interfaces can scare users away, even if the underlying application is great.
Then I would recommend attempting yourself at small real projects. Like a small game, a website or another project that has use, but is not too big that you will be lost in its large size. Re-implementing existing applications and/or games can be a great idea to train your programming skills.
Specifically game development training programmes exists, but I do not know any. But neither do I truly believe they are necessary. If you know your way around a computer, most tasks should be relatively easy to overcome. While you may have to learn new environments, etc., even that should be familiar in some way or another.
So before you start at a university, I will recommend trying yourself at small projects, in simple straightforward programming languages, such as Python.