You have a lot of options and it depends on your spiritual likes and dislikes.

For the limited time you have (I would add 1-2 weeks if possible) the following is the classical spiritual trip itinerary for the south that focuses in Chennai and around in the state of Tamil Nadu -

Tiruvannamalai: Arunachala, Ramana Maharshi and many many others who are there in this seasonal spiritual supermarket, see http://www.gurusfeet.com/guru/ramana-maharshi#tabset-4, a must!

Aurovile and Pondicheri: The remarkeable meditation sphere and the community in Auroville and Sri Aurobindo and the french colony in Pondichery, see http://www.gurusfeet.com/guru/sri-aurobindo

And around also: Mamallapuram, Kodaikanal, the Tibetian settelments, Puttaparthi (Sai baba).

In Kerala you have also the ashram of Ammachi, the hugging mother and in Puna you have the resort/ashram of Osho but these are far away and you better avoid long travels if you have only 1-2 weeks.

See here in the guru profiles' locations tabs - there is a lot of other valuable information...

Traveling to India is an unimaginable experience and fun for a spiritual person.