I will be going to south india in November this year..
sharing my raw plan.. 1.delhi to Bhubaneswar
2.bhubhaneshwar to chennai
3.chennai to tirupati
4.tirupati to madurai
5.madurai to rameshwaram
6.rameshwaram to kanyakumari
7. kanyakumari to Trivandrum
8.trivandrum to delhi Total days - 11
All hotel ,flight,train and darshan booking done
In Karnataka try Hampi, Badami, Aihole and Pattadakkal which is of historical importance. And for trekking, the Kodachadri trek is suitable for the month of November. please read my blogs on two of these spots from my profile for more information.
In Kerala idukki and alappuzha is the best spot you can visit