If you are reliant on public transport, and if you are already in Kerala, the most economical way to get there would be by bus from Kottayam. You should be able to find a bus going to Thekkady in the bus station easily. It should take you about 4-5 hours.
If you are travelling from Tamil Nadu into Thekkady, then the best way to go about would be to bus it from Madurai or Dindugal. Madurai is well-connected by road, rail, and it has an airport as well.
It was supposed to be a stright forward bus ride to Kumily... But like many indian bus rides something unexpected happend, and we ended up in a jeep!
At 6:30 am one morning (in 2005) we went from Munnar in a bus, we had the front seats and where both happy and unhappy about it. We had great views, but it was a bit scary because of the narrow road...
It turned out that the bus did not go to Kumily after all, but we could ride with to Pooparai (?) and change bus to Kumily there. But when we came to Pooparai it turned out that it was many hours to the next bus would depart. So we rented a jeep-taxi for the rest of the way to Kumily. The driver drove like crazy, on a road that was winding trough the beautiful green landscape.