Hampi (Off beat place)
Gokarna/Pondicherry/Varkala(if you want to go to beaches)
Ooty/Coorg/Munnar (If you'd like to visit a hill station)
Bangalore (for city life with good weather)
Kanyakumari (Southern tip) Hope this helped? if yes, please upvote.
Bangalore- You can also opt for exploring some nearby places like-Mysore, Coorg,Chickmanglur. You can also plan a trip to Tamil Nadu where in you can go to Ooty, Coonor, kodaikanal.
Rameshwaram and kanyakumari is also an option. kerala- Go to Kochi- from there you can visit Kerala backwaters, Allapuzey beach, Varkala(2hours from Allapuzey beach).
The the the best state is Kerala.... beautiful destinations at cheap prices...but as u have asked for cities in South India...here are some Kochi
Pondichery aka Puduchery