There is a domestic airport in Rameshwaram. But, it is not connected with most of the cities. If you are planning to use airways, you can fly to Madurai. From Madurai, there are a lot of flights to Rameshwaram. Madurai is about 163 km away from Rameshwaram. The nearest international airport to Rameshwaram is in Chennai and Coimbatore.
If you are planning to choose railways, you can get to Rameshwaram via Madurai. Madurai is connected to most of the cities in India through railways. The same goes for buses too. Coimbatore, Madurai and a lot of cities in South India are connected to Rameshwaram through bus route. Though there are a few buses that connect North Indian states to Rameshwaram, the frequency of those buses is very less.
The city is well connected with major cities in India through highways. Thus, it is possible to self-drive or opt for tourist cabs to reach Rameshwaram. The Indira Gandhi Bridge connects the island to the mainland of the country. Thus, most of the road and rail transportations are easier.
Inside the town, you can hire rickshaws or jeeps to travel around.