Game Design and Development

For game designing, you can pursue animation and VFX. I think for game designing, Arena animation is the best academy.
For game developing, you can check Maya academy. Both are best, but note that designing and development are different field of gaming industry.

There are two fields in the game industry:
1) Designing: which include animations.
2) Development: Which is pure coding and testing part.
As you have mentioned development, I think you have to visit MAYA academy near to your location which offers various course in game development part time and full time and it is also a good academy as per my knowledge.
For designing, Arena animation is best in the industry.
All the best.

You may learn game designing for free from eDX or Udemy.

You may do the course from the below link:
https://www.edx.org/course/introduction-game-design-mitx-11-126x-0
You may also refer to websites like coursera and udemy for the same.

Refer to the linke below:
https://animation.shiksha.com/game-development-courses-in-pune-categorypage-12-94-1-0-0-174-114-2-0-none-1-0

These are the best institutes for animation and gaming courses:
1. Frameboxx Animation and Visual Effects (Frameboxx)
2. Arena Animation Academy
3. Maya Academy of Advanced Cinematics