The city centre — palace, market, art gallery — is compact enough to walk. Auto-rickshaws are the standard way to get around and are metered, though the meter is not always used; agree a price before getting in for any journey of more than a kilometre or two. Mysore has a well-spread city bus network run by KSRTC that covers the outer areas including Chamundi Hill.
Renting a bicycle is practical given the flat terrain and relatively light traffic. Several guesthouses in the Sayyaji Rao Road area rent cycles by the day. Hire a car and driver for day trips to Brindavan Gardens, Srirangapatna, or the Bandipur forest — taxi aggregators and hotel desks can arrange this easily.