You can take a direct flight to Madurai or Chennai
I booked an Air India Premium Economy from Delhi to Chennai
Metro from Chennai Airport To Egmore Station
From Egmore, there are few overnight direct trains To Rameshwaram
I avoid Cabs for long journey so prefer trains
Otherwise you can take a direct flight to Madurai from Delhi, Mumbai, Bengaluru, Hyderabad etc and take a cab/bus from Madurai to Rameshwaram (this is better option)