Mantralaya is around 250 km from Hyderabad, the journey from Hyderabad takes about 6-8 hours by road. It takes about 10 Hours by road from Bangalore. If you are driving from Hyderabad, you need to take the NH7 - the highway to Bangalore and take a diversion at Jadcherla. Various state roads will take you to Mahboobnagar and Raichur in Karnataka.
Mantralaya is around 600 km from Chennai. It can be reached through the route of Chittor. It is around 650 km from Mangalore. By road, one can take private buses from Mangalore to Bellary and from Bellary one can get ample number of KSRTC buses to Mantralaya. The journey takes almost 15 hours.
The nearest rail head is Mantralaya Road. It lies on the Mumbai-Bangalore, Delhi-Bangalore and the Mumbai-Chennai train routes. Regular rail service is available for this route. Apparently, many trains like Karnataka Express, Rajdhani, etc. stop at Mantralaya Road to refill water, even though the station is not on their official list of stops. It may not be a good idea to rely on this information though.
There is one train that goes from Hyderabad Deccan station to Mantralaya Road en routeTirupati. Mantralaya Road is about 16 km away from Mantralaya. You can travel to Mantralaya from Mantralaya Road station by private Jeeps, taxis, autorickshaws or public bus.