Bicycles are how everyone gets around. The flat coast road from town south to Manzanillo is 13 km long with no hills, no traffic to speak of, and the beaches strung along it. Dozens of shops in town rent bikes by the day for a few dollars; check the brakes and bring a lock.
For longer trips, shared taxis and collectivo vans ply the road south every half hour or so; flag them down anywhere. Conventional taxis wait at the bus terminal in town, and most hotels can arrange a private driver to the Manzanillo end of the coast or up to Cahuita. The bus from Puerto Viejo to Manzanillo runs several times a day and is the cheapest option of all.