Game design

Game designers obviously focus on designing games. This is quite different in terms of the work vs. animators who usually work on movies or commercials. Whether it be game design or animation, there are many specific skills to learn and to concentrate in. You have the modeling of the characters (which is akin to digital sculpture), character animation, which requires good fine art skills and drawing of faces, environmental design, which is essentially designing the background for the characters, texturing (for skin and fur), lighting, rigging (to get the character to move correctly) and more.
I guess the bottom line is that:
1. You need to decide whether you want the programming side or the design side.
2. If design, majoring in either game design or animation would probably be just as good for developing the above-mentioned design skills. However, if you want character animation or environmental design, it is crucial that you develop your drawing and illustration skills particularly of characters. Other aspects of animation don't need this as much.
3Ds Max
3Ds Max offers better interoperability with other Autodesk products.
Many prefer 3Ds Max for materials and rendering due to the ease of materials setup.
Getting the most out of 3Ds Max can require some scripting using MaxScript.

According to me, Game design and development is much better as compare to 3DS Max, because by learning any software you can't give great output, for a great output, you have to work on your things in some manner and you should decide some goal so for that you can choose Game design and development.
There is a vast difference between learning a tool and learn to using that tool for your own work, if you will learn how to use any tool for your own work, you will be able to complete your work in a proper manner or if you are focusing only tool, then you will not be able to work beyond that. By learning only 3DS Max, you are making your own limits. But, to survive or to create something you should learn how to use 3DS Max or any other software for your own concept.
So, go through Game Design and Development and learn the way to use tools, not only the tool by that you can execute your own ideas and thoughts. And by learning 3DS Max, you can do so many work, but that also will be limited, because you will not be able to draw the things in that and you will not be able to make any video and present your own thoughts.

India or anywhere else, the procedure to follow is the same. Remember contrary to other fields in IT this is one of the most challenging fields.
Best way is to start with computer science subjects with key courses in C++, Object Oriented programming and most importantly Algorithms and Data Structure.
Once done choose a full fledged game engine to work with and spent a few days learning the native API. This will be if you understand Object Oriented programming.
Once you have learned the API learn about the different pipelines in game design such as I)physics ii) gameplay and control ii) AI iv) User interface v.)particle effects etc.
Once done move to advanced topics of writing shaders, optimization, parallelism, texture baking etc. You will know basics each of these as you start learning.
This are content for the programming and technical side.
Second is the art pipeline and this requires you to be a native artist I.e. good in drawing. According to my experience in the art part if you can draw you can design anything without much efforts. Just following the steps in help menu of any 3D modelling and animation tool such as max maya or even photoshop and illustator for 2D art.
If you are not an artist you can take help or OpenGameArt.org can help you to get art assets for your game.
Build your ides and publish a few games to get known among the community. Now you will be good to go to work with a professional studios such as EA Ubisoft or you could start a indie game development studio of your own.
As you will dive to the deepest level of game design instead of designing games you will be designing tools on which games are built

Game design and development is the course in which a student learns about how to design, develop a digital prototype using an interactive design process and language. There are some good colleges which provide certificate course or Diploma course in this and also a Bachelor's in game designing and here are a few colleges:
1. Backstage Pass
This college offers B.Tech in computer science specialisation in gaming design. It also offers BFA game art and design. This is one of the best designing colleges in 2010 in India situated in Hyderabad.
http://www.animationcoursesahmedabad.com/
2. IIFA Multimedia Srinivasanagar
This college offers B.Sc in Gaming Designing. This is one of the best college est. in 1991 affiliated to Bharathiar University. This college offers placement to students into various MNCs.
http://www.iifamultimedia.in/
3. Maya Academy of Advanced Cinematics, Shivranjani
This college offers Diploma and Bachelors course in VFX and game designing. This college has the best faculty and good learning environment which gives good 3D learning and designing course to the students.
http://www.animationcoursesahmedabad.com/

It is really tough to directly comment on any institute is best or not or answer which is the best institute in a city or a country, especially for domains like web designing, animation and VFX or gaming, etc., where many institutes offer many types of courses for a different duration for the same pipeline.
We need to see the content of the course, duration, cost, infra, earlier students outputs and skill developed, in-time course completion record, certification process & delivery of the certificate in time, etc.
Also, the same brand has different centres and branches, and fortunately/unfortunately all does not perform the same. Some centres are doing great, some not. It varies from place to place, centres to centres and also on the local management & team.
So, it's always good to visit the institute personally, speak to them, see the track record and then decide if it's up to the mark as per your requirement.
The user recommends following institute:
Aquarius Animations Pvt. Ltd.