Auto-rickshaws are the standard way to get around Jaipur. Always agree on a fare before getting in, or insist on the meter -- expect to pay 50-150 INR for trips within the city. Cycle rickshaws are cheaper and fine for short distances in the old city. Ride-hailing apps like Ola and Uber work well here and are often the easiest option.
The old walled city is best explored on foot, though the traffic can be intense. Jaipur also has a metro line connecting Mansarovar to Badi Chaupar (useful for getting to the old city) and a network of city buses, though both are more practical for locals than tourists.