The old town is best explored on foot — the historic centre is small enough to walk across in twenty minutes, and the network of narrow shophouse-lined streets only really works at walking pace. Most of what you came to see is within this compact area.
For longer trips, Grab (the Southeast Asian Uber equivalent) is cheap, reliable and the standard way visitors get around. A typical trip within the city costs RM 5-15. Download the app before you arrive and add a payment method.
The free CAT bus loops around the old town tourist area roughly every 30 minutes — useful for footsore moments. The wider Rapid Penang bus network covers the rest of the island, including the route to Kek Lok Si and Penang Hill (Bus 204 from Komtar). Buses are slow but cheap; pay cash on board.
Trishaws (cycle-rickshaws) operate around the old town. They are a tourist experience rather than transport; agree the price before you set off and treat the ride as a sightseeing tour at trishaw speed.
Bicycles can be rented at several places along Chulia Street and at Hin Bus Depot. The flat old-town streets are mostly fine for cycling, though traffic on the larger roads is intense and not bike-friendly.
To reach the mainland, drive across the bridges or take the Penang ferry from Pengkalan Weld.