only ten days

You should visit Rajasthan, Kerala, Goa

I would recommend going to Manali, Spiti Valley, and then Leh . All these places are ideal for treks and biking. I just came back doing the same things. :)

You can visit Mumbai, Udaipur, Jodhpur, Pushkar, Jaipur, Mumbai during your tour. 

I am late here to answer but plan for Rajasthan. Rajasthan has rich culture, heritage, history, palaces, wonderful food, forts, temples, architecture and most important helping people. You can visit below cities in Rajasthan. 

Jaipur

Jaipur, Jodhpur, Jaisalmer, Udaipur, Ajmer, Pushkar, Mount Abu

Below are some other places you don't want to miss.

Ranthambore Tiger Reserve, Keoladeo National Park

I suggest you guys to go to Rajasthan is a state not to far from Mumbai. Taking a train in India is in an adventure, but it is functional. If you rather stay in around Mumbai, go to Pune and then go down to Goa. If you decide to go to Rajasthan, go to Karni Mata Temple, Udaipur, Jodhpur, and Jaipur. 

Or, you could take a flight to Amritsar to visit the Harmandir Sahiband to travel around the North area like McLeod Ganj and Manali. These are really nice places to trek and to meet cool people. 

I'd try to get out of congested, dirty and stressful Mumbai as quickly as possible and head somewhere else.  If you want to head south, then Goa is a nice beach city, Pune is a refreshing highland town, and Kochi is a large city in southern India with plenty of historical attractions, including several related to the history of India's Jewish community.

If you want to trek, then Leh and Dharamsala (home-in-exile to the Dalai Lama) are great bases for trips into the Himalayas.

If you're into a more spiritual trip, then Amritsar (the Sikh holy city) and Varanasi (a Hindu holy city on the banks of the Ganges) are worthwhile spots.

Corbett would be a must. Also, do visit the west coast of Indian (which is called Konkan). There are numerous treks in that area. I think these two things would be more than enough for 10 days. :-)