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Winter: Tamil Nadu does not have extreme weather conditions and hence can be visited anytime during the year but winter months can be the best if one wants to experience pleasant weather while travelling in plains. You can visit tourist attractions like Thanjavur and Kumbakonam during winter months.
The highlights of visiting in winter is that you would get to experience the celebrations of Pongal, the Harvest Festival in January and hence can be said to be the best season to visit Tamil Nadu. It is the festival of Tamil Nadu celebrating the farmers who provide food and representing the culture of the state.
Summer: Summer weather in Tamil Nadu is very hot but not extremely hot like North Indian states. These are the times you can visit hill stations like Ooty and Coonoor in Tamil Nadu.
Monsoon: Though the monsoon season is from June to Septemeber, June months are generally a bit hot and July to August are breezy and pleasant with occasional rainfall. The peak monsoon months are July & August. Unless it is a heavy monsoon, monsoon months are quite good for travelling to Tamil Nadu. September is called the second season in Ooty and hill stations receive heavy footfalls in September as well.
This said, the best months to visit Tamil Nadu are winter months of November to February, October and March are mild, it is best to avoid visiting in the summer & monsoon months of April to September.