The nearest big towns (an hour by flight) in south from Mumbai (Bombay) would be Bangalore (Hampi is close to Goa and is a must see, also known as the rome of the east) and then Hyderabad which is the fifth biggest city of India (After Calcutta, Bombay, Delhi & Madras). There are two places closer to Mumbai (excellent for trekking), one is Lonavala (almost like a hill station, half way between Mumbai and Pune) and Mahabaleshwar (after Panchgani) which is about 2/3 hours by road from Pune and then there is Ajanta and Ellora caves near Aurangabad city.
South India is popular for its temples, some beaches (very unlike Goa) and is relatively cleaner and richer than the north. South indian breakfast is very popular in the whole of India.
Personally I would prefer to go north as it has so much to offer. Rajasthan has a lot to offer...palaces and forts and also one of those train rides called Palace on Wheels. Miles and miles of musturd fields, a lot of camels...there are some beautiful towns in Rajasthan like the Pink City of Jaipur (if in Jaipur, must see Chowki Dhani), Jodhpur, Udaipur, Jaisalmer etc.
Although born in the East near Calcutta, I lived in Hyderabad (south India) all my life and have been working and living in the west for the past 11 years (US and UK) so I am baised when I say South - but I like north as it is much cooler. The food is similar to what you eat in curry places in England.
Here's a link that might also help:
http://www.incredibleindia.org/