Weather wise October to February is the generally good season to travel in our country. This season one can travel in any part of the country. This is because the monsoons would have subsided by October. The NE monsoon may start in the eastern coast states. From November onwards the major parts of the country will start experiencing pleasant cool weather and some parts will have extreme cool weather during December to early February. But cool weather is more tolerable with proper clothing than the extreme hot weather conditions in the March to May end season. Summer season travel is more suitable to hill stations and the states of Kerala,Karnataka and the hill stations in the northern and southern states.
However people choose to have different purposes and enjoyment. For example, the monsoon season of June to September is suitable for those who enjoy the heavy rains and full flowing rivers and lakes in Goa, Karnatak, Maharahtra, Kerala . The popular water falls like Kuttalam and those in the Nagercoil, Tirunelveli belt can be best visited during the god rains in Kerala. Similarly there are many other water falls in TN, Karnataka also.
January-February is good season to visit Munnar in Kerala. That can be combined with the backwater tour staying in houseboats etc. The period January onwards is the season of many local festivals in the country and those who like to see the local culture and traditional habits and local cuisine can use this season for touring.
Those who have not been much familiar with water bodies can take a tour of Kerala, Karnataka and Tamil Nadu where you can have the hills, lakes, rivers and sea. The house boat stay in Kerala will be good for viewing live fishing and cooking fresh caught fish, for those who relish that. The hills of TN, Kerala and Karnataka have the spices cultivation and tea and coffee plantations.
Indian Railways have their packages in different categories. IRCTC announces such packages periodically.
You may not be able to cover all states in one stretch of a tour, but can visit a few states n one package. As of now, your tout destinations can be the hill stations like in Northern states and North East state and a few places like Ooty, Kodaikanal,Yercaud etc in Tamil Nadu. But if one can tolerate the hot weather then relatively the four southern states except a few parts of Andhra –Telegana are okay. Maharashtra and Goa also can be toured this season.