Whoa - that's a really nice trip with interesting options.
There are direct flights and buses but no direct trains for this journey. Catching a direct flight is the fastest way to do this journey. Also direct buses are available. You can also travel there by taxi. Hyderabad (Andhra Pradesh) to Goa (Goa) distance is 661 km and you can get there by flight in just over 1 hour and 30 minutes. Going by train + train won't be that fast but you will get fares starting from ₹ 434.
Beautiful weather for sightseeing and other outdoor activity.