Hyderabad is vast and distances are long, so some form of transport is essential. The Hyderabad Metro is clean, cheap, and covers the key corridors between Secunderabad, the Old City (Sultan Bazaar / MG Bus Station), HITEC City, and the airport. Auto-rickshaws are everywhere and metered in theory; in practice agree a price before setting off, or use Ola or Uber which both work well here.
For the Old City, walk. The streets around Charminar, Mecca Masjid, and Laad Bazaar are impossible to drive through anyway and much more rewarding on foot. Keep the afternoons for the air-conditioned museums and restaurants — midday in Hyderabad is not a time to be out on the pavement.