The town itself is compact and best explored on foot. Suggestions in guidebooks that Ban Santhichorn and Lisu Village might be reached on foot are optimistic.
For exploring further afield, bicycles (40-100 baht/day) and motorbikes (from as little as 80 baht/day) can be rented from many agents along the main street. The roads around Pai are steep and obtaining a decent mountain bike with fully functioning gears is surprisingly difficult. A motorbike is definitely the better option if you can drive one. aYa Service in the town centre rents motorbikes for 100 baht (and a 100 baht helmet deposit), plus 40 baht for damage insurance, 40 baht for theft insurance (passport taken as deposit). You also have the option to return the bike in Chiang Mai.
You'll also want a motorbike if you're planning on staying in some of the outlying bungalows in the valley around the town.
Motorbike and 4WD taxis are also readily available.