Pack some nice homemade food, cause this isn't a short trip !
There are direct flights and trains for this journey. New Delhi (Delhi) to Bhubaneshwar (Orissa) distance is 1742 km and you can get there by flight in just over 1 hour and 55 minutes or by train fares starting from ₹ 547.
The heat there will make you sweat, but it's surely bearable.