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Hi Vamshi,
I would be happy to understand your current stage of progress including academics, courses, hobbies and other interests to help you understand what in you can lead to game design and also what aspect of game design can suit you better. In most cases a broader learning like a combination of graphics, multimedia, web design, VFX, 2d and 3d animation, can lead to a better preparation towards launching oneself into a gaming career.
A specialization in Game Design is a useful add-on afterwards. Just to state the facts, none of the famous game designers in the world today have done anything remotely linked to game design. That said, do not take it as a bar to entry, but to elucidate the fact that a broad learning is always necessary to get into a challenging career such as game design.
Now let's go into the nitty gritty on Game Design. To become a Game Designer, you must get a good knowledge of 3D software as well as preproduction. Games environment is created in 3DSMax and Character animations in Maya. Only the models are made with less polygons and more textures. So, take a program that teaches you these. It won’t matter whether the program is called game design program or not. Rest, to become a level designer you must have a passion for game design. Also you must have passion for playing all kinds of games including card games, dice games, sports, etc.
The 3rd option is Game Art (it's different from Game Designer), where you design the game itself, i.e. what happens on each level, and how things go about as well as the base story of the game. It is a stage that you reach due to your vast knowledge and passion and even exceptional technical skills cannot help you reach. This is usually the story designer. May I ask, have you ever designed a game? Finally, to do game programming you would require knowledge of programming in which usually a B.Tech/MCA would help.
Typically, game programmers usually would know C#, Java, besides at least a Game Engine. Here is a complete snapshot of Gaming Industry by me on Shiksha website:
https://www.shiksha.com/from-gaming-to-game-designing-article-5827-1
Wish you the best for your career. If you like this answer you may select it as best answer.