Auto-rickshaws are the main way to get around Gorakhpur; agree on the fare before you get in. Cycle-rickshaws handle the shorter hops in the older parts of the city. App-based taxis (Ola, Rapido) have a reasonable presence and remove the haggling.
The city is sprawling — the railway station, Gorakhnath Temple, and Ramgarh Tal are all in different directions — so walking between major sights is impractical. Budget an hour's travel time for any cross-city journey.