Animation

You said you want to become a game developer. Is it game programming you are referring to? A game developer can be any person involved in the development of the game. He can be a programmer, producer, artist, modeller, etc. I am gonna assume you are interested in the programming.
If programming is what you want to do then I suggest you take up any Bachelor's degree in the field of computer science or IT (information technology) and learn about computers and programming at the college/university. When at home start implementing your skills by making games. There are plenty of resources available online which teach you the ways to begin with the game programming. I recommend Unity3D as a beginner game engine as it is easy to begin with and there are courses in Udemy for free which will cover basics. You need to learn C+ in order to code with Unity3D. However, if you find programming a bit difficult in the beginning you can look at construct 2 game engine to understand the basic flow of the game as it requires no coding. You can find tutorials to get started with the contruct 2. The only way you can learn game development is by making games so try to make as many small games as you can while you are at college. Once you complete your education, you can start applying in companies by showcasing the games you made.
In the game industry, your portfolio carries more weight than your degree. The degree here helps you in communicating with computers in a better way. So make sure that you are focused more on your goals.