Hello! I am not a game developer by trade, nor am I in the EU so I'm unfamiliar with the specifics of your school system, so this advice will be more general.
As you seem to have realized, the first thing you have to decide is what exactly you mean by "game designer." Writing a game is a complicated, usually multi-person process, with different specialties involved (I talked about this a little here: https://careervillage.org/questions/1885/do-i-have-to-be-good-at-art-to-work-in-a-game-company ).
You would likely only need to be particularly skilled at both art and programming if you try to make a game with a very very small group of people, or possibly just on your own, which is generally called being an independent (or "indie") game developer.
Unfortunately I cannot give much useful advice on if you focus on art, but I can tell you about programming.
Programming makes use of a lot of concepts from Computer Science (they are not technically the same, though I got a degree in Computer Science in order to be a programmer), so any classes relating to that can be helpful as well. They are usually college-level courses in the USA (so likely to be taught when you are 18 years old or older in our school systems).
In my personal experience, the things you do on the side are equally important to the classes, if not more, for becoming a game programmer, if that is what you choose. Really, if that is the direction, you want to become a programmer in general, and most of the programs you write can be games.
The great thing about learning to be a programmer is that it doesn't require you to be learning it in school, and while it does require a computer, once you have access to one, particular with an Internet connection, you can start to gain the skill on your own. There are websites, like http://www.codecademy.com/ , where you can start to teach yourself to program. You might not see much of a connection between "Write a loop that counts from 1 to 10" and creating, say, a video game where penguins swim around, catching fish, but all the skills are layered. After all, the way the computer would move each penguin in turn, in such a game, would be a very similar loop to "count from 1 to 10".
I apologize in return that this was an even longer answer. In short, though, determine what part of game design you do want to focus in, and if it's programming, start learning on your own. This is likely true for art as well, come to think of it.