Hi Mayank! The cheapest way to get to Goa from Bhubaneswar is via train.
The Amravati Express is a train that goes directly from Bhubaneswar to Goa. It runs in Tuesday, Wednesday, Friday and Sunday. The travel time is approximately 32 hours and leaves Bhuwneshar at 6:10 am in the morning. It arrives in at Madgaon Junction in Goa.
Happy Travelling!
The fastest train from Bhubaneswar to Goa is 18047 Amaravathi Exp. It takes about 31hr 45min hours to reach Goa. This train leaves Bhubaneswar (LGTR) at 06:10:00 and reaches Goa (MAO) at 13:55:00. It operates on Mon, Tues, Thur, Sat.