Hi Ameer, here is a list of places you can visit while taking this trip
Undavalli Caves(Andhra Pradesh)
Chandragiri(Andhra Pradesh)
Hampi(Karnataka)
Badami(Karnataka)
Pattadakal Group of monuments(Karnataka)
Aihole(Karnataka)
The Chola Temples(Tamil Nadu)
Thanjavur(Tamil Nadu)
Monuments of Mahabalipuram(Tamil Nadu)
The cost would depend upon how are you travelling and where you are staying. You will have to plan a map from where you are starting your journey and cover every destination in one state then move on. Since Dravidian culture was spread across in whole of south this is going to be a long journey.
Visit thanjavur and places around it. I'm sure you will it extraordinary if you're a history buff. 100's of temples are there which explains the cultural heritage of chozhala kingdom. consider gangaikonda chozhapuram, kumbakonam, thiruvarur, poombukar, nagapatinam, vedaranyam to seek out some amazing sculptural history