Sayulita is small enough to walk entirely, and most people do. The central streets are cobblestone — fine for sandals, uncomfortable in heels — and the main beach, plaza, and restaurant strip are all within ten minutes of each other on foot.
For the beaches outside town, mototaxis (small motorcycles with passenger seats) wait near the plaza and will take you to Playa de los Muertos or the north beaches for a few pesos. They're the standard way to get around when you don't want to walk in the heat.
Renting a bicycle works well for exploring the town and the nearby road to the highway, but the terrain immediately around Sayulita is hilly enough that it's genuinely tiring in warm weather. Golf carts are available for rent and are common among families staying in the hillside villas.
For day trips to Puerto Vallarta or Punta Mita, shared vans (colectivos) run from the highway junction north of town, and taxis are available at the plaza for the full trip. The airport at Puerto Vallarta is about 40 minutes south in light traffic.