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Sharath,
I'd be happy to understand your current stage of progress including academics, courses, hobbies & other interests to help you understand what in you can lead to game design & 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 & 3d animation, can lead to a better preparation towards launching oneself into a gaming career. A specialization in Game Design is a useful addon 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 & Character animations in Maya, Only the models are made with less polygons & 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 & 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 be knowing C#, Java, besides at least a Game Engine. Arena offers a very good Course in this regard namely AAIP where you can learn enough 3D Art and Effects for Gaming.
I wish you the best for your career. If you like this answer, you may select it as the best answer.

Riya,
I am not sure how good you are with Maths and programming. If you are, then you must pick up a language whether PHP/ HTML5 (web dev), .Net (for software development), Java/ Android (for apps) or C#/ Unity (for games).
However, like I mentioned, you must be good with programming to move ahead in this area since you have to compete with other MCAs and B.Tech (CS) applicants. So, choose your internship/ industrial training very carefully. Do not just go for the cheapest to save money. See the IT Company's credentials on the net. For example, if someone provides LIVE PROJECTS, are they there near the TOP on PORTALS like JustDial? Or, if the company is registered with STPI (Software Technology Parks of India) - all good IT companies are registered with STPI. If a company spends money to market themselves to customers, only then they are able to get LIVE PROJECTS.
Also, if you are an average programmer, and if you you're better with creative work, you can head for web design or even VFX/ Animation since there is a huge requirement coming up in that domain. Games Design and Games Programming/ Development.
Wish you the best for your career. Remember, your Career is a resultant of your knowledge combined with your passion and expression ability. If you like this answer, you may select it as the best answer or press like.