Whoa - that's a really nice trip with interesting options.
There are direct flights and buses but no direct trains for this journey. Hyderabad (Andhra Pradesh) to Goa (Goa) distance is 662 km and you can get there by flight in just over 1 hour and 30 minutes or by train + train fares starting from ₹ 434.
The heat there will make you sweat, but it's surely bearable.