The journey from Gangtok to Madurai covers roughly 2800 km. You must first reach NJP or Bagdogra. From NJP, you can take a long-distance train to Madurai Junction. Flights from Bagdogra to Madurai usually connect through Chennai or Bangalore.
Total Distance: 2800 km by road, 2300 km air distance.
This is a very long journey; ensure you have a confirmed berth.
Chennai is the most convenient hub for this connection.
October to March for pleasant weather.