Game design

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.

There are three most common pathways to enter the game industry: as a programmer, as an artist, or as a tester. I wouldn't really recommend the third route.
1. Get a degree
The first step is to get a four-year degree. If you're interested in programming, get a Computer Science Engineering degree. If you're interested in art, get a degree in Fine Arts. If you are into writing, get a degree in English / Literature. (Some other distant options are starting out as a game writer, for which a degree in mass media / journalism can help.) Do not go for diplomas / degrees in animation / game design. They are usually expensive and generally not worth the money that you put in.
2. Intern during summers
The best way to learn fast is by joining a small game studio or company that is looking for interns. It can be a great experience. Or do your own game project during the summer. Gather up some teammates brainstorm and start prototyping an idea. Or go solo and build a game during a weekend.
3. Connect with people & attend conferences
The best way you can socialize with people in the gaming community is by being a part of gaming related forums on the internet and attending conferences. Watch out for local game dev meetups in your city. Announcements for such meetups are usually made NASSCOM Gaming Forum.
Some conferences that will be of interest are:
NASSCOM Game Developers Conference (Pune, November each year)
Casual Connect Asia (Singapore, May each year)

Below we have listed top FIVE Gaming & Animation institutes in India, please note they are listed not based on the rank.
AIGA Bangalore - Asian Institute of Gaming and Animation
AIGA started in early 2008, claims to be the first dedicated Game education institute in India, in initiative of Edulink India Pvt. Ltd. offers various Certificate and Diploma courses in Game Art, Design & Programming.
For more details, email AIGA: admissions@aiga.in or visit their website http://www.aiga.in/
Backstage Pass school of gaming, Hyderabad
Backstage Pass fulfills your dream to become Professional Game Designers, Game Artists, Game Programmers, Animators, Game Producers or Animation film Producers through their career focused graduate and certificate programs.
For more details, email: information@backstagepass.co.in or visit their website http://www.backstagepass.co.in/
IACG- International Academy of Computer Graphics
IACG, established in the year 2004, claims to be India's first institute to offer Bachelor's and Master's programs in Multimedia & Animation in collaboration with Jawaharlal Nehru Technological University(JNTU), Hyderabad.
For more details, email: contactus@iacg.information or visit their website http://www.iacg.co.in/