Kolkata-Vienna-Innsbruck-Prague-Budapest-Vienna

Skip Innsbruck, unless you have pressing reasons to go there. Vienna, Budapest and Prague are major cities, each of which needs at least the three days you've got.
The best way to tackle this trip would be to have two nights in Vienna on arrival. Take the train to Budapest. Have three nights. Take the train to Prague via Vienna. Have three nights. Return to Vienna and have the remaining nights before taking your return flight.
Advance booking of your train tickets on http://fahrplan.oebb.at/bin/query.exe/dn can save you considerable money.

tonfromleiden is giving you good advice. You land in Vienna so spend three days there. Then go by train to Budapest and spend 3 days there. Then from Budapest by train to Prague (about 7 hours). Direct train from Budapest to Prague does not go via Vienna. It goes via Bratislava and Brno. Then spend three days in Prague and fly home from there. Skip Innsbruck.

Thank you so much!! But I would definitely want to do Innsbruck as nature is at it best in this place. Would you then want me to drop one of the cities and go for Innsbruck?

I'm not so sure about nature around Innsbruck, but if you want another city in exchange, I'd say Budapest. Vienna and Prague are absolute musts.

If you want nature and the alps, I would try something other than Innsbruck. I'd go for the Wolfgangsee area, for example. You only have time for 3 destinations. 4 is too many.

The best of the three is obviously Budapest, but because Budapest is so worth the time I would skip it until you have at least 4 nights (3 full days) and 5 nights is better. You can do both Prague and Vienna in 3 nights each. Or as an option start in Budapest and if you discover that I am correct you can amend the trip on the fly.

Hi dipannita. Good idea to not spend your whole trip in cities. Austria has beautiful mountains, very good idea to spend a few days of your trip among them. I would drop one of the cities so you have time for the Austria mountain areas. Enjoy your trip!