Whoa - that looks like an exciting journey to take !
There are direct flights and trains for this journey. Travelling by train is the cheapest way of getting there. New Delhi (Delhi) to Goa (Goa) distance is 2017 km and you can get there by train in just over 29 hours and 45 minutes. However, flight tickets come also cheap with fares starting from ₹ 4600.
The weather is amazing right now. You'll have a great time.