How long should you go for:
I would say a minimum of 6 days should be considered if you're looking at two cities. Anything less than 3 days in each city would do justice to either. Since Russia is such a massive country, and remains so little explored, I would recommend adding Sochi to your trip with an additional two days if possible.
Where to Stay:
In St. Petersburg, you don't need to worry too much about the location of your hotel, as long as there's a metro station within walking distance. It's a small city and the best thing to do is to pick a hotel that offers a shuttle service to the city center. I stayed at the Park Inn Prebaltiskaya, and the hotel was one of the better deluxe (4 star) hotels, apart from the supplement of 300 roubles they charge for in-room delivery. In Moscow, it's best to book a hotel closer to the city center, or one that's within 3 or 4 metro stops. This will ensure you're close to the heart of the city, since it's huge and could easily cost you an hour or so in commute. Given that the thriving nightlife of this city is worth exploring, it makes little sense to book a hotel that would keep you from venturing out too late at night.
Azimut Olympic is the ideal hotel for an Indian traveler. It isn't expensive for a 4 star hotel, yet hosts an Indoor swimming pool as well as an Indian restaurant. It's within comfortable distance from the main sight-seeings as well. Another one I recommend is Izmailovo Delta, though it's considerably far from the main areas worth exploring. The rooms are comfortable and clean, the hotel is walking distance from the metro and a couple of fast-food restaurants. If you're on a lower budget, this hotel is ideal.
What to do:
There's plenty to do in Moscow and St. Petersburg to keep you occupied for 3 full days! While Petersburg doesn't have much to offer for children beside it's zoo, there are enough options to pick from in Moscow. Whatever your age and interests, something is sure to click. A Neva river cruise, visit to the Hermitage and Catherine's Palace should be at the top of your bucket list, right alongside watching a ballet performance at the Bolshoi Theatre in Moscow and dining at Cafe Pushkin! Check out: 12 Things You Must Do in Moscow