If you consider recreation, especially with children, Mumbai (Bombay for the "old style") is not the best option. Industrial business city, with a fast pace of life similar to Moscow. Yes, and the beaches are very dirty.
To see India as such, I advise you to choose a rich and interesting program Golden Triangle (visiting cities: Delhi - Agri - Jaipur), which can be combined with rest in Goa. But again, it is more suitable for adults who like to travel, for children this tour can be tiring.
The most suitable place for rest with children is South Goa, where more hotels of 4/5 level are concentrated.