From the airport in San Jose, you can take a taxi (available in front of the airport, once you leave baggage claim) to the bus depot in San Jose. From there, you can take a bus to Puntarenas. The trip will take a couple hours depending on traffic and weather conditions, but its inexpensive.
Local buses leave from downtown and are relatively inexpensive. For travel to places outside of Puntarenas (ie, Jaco, Manuel Antonio, etc) there's a main bus station. Buses are usually on time and sometimes even leave early, so plan accordingly. Don't be surprised to see vendors on buses selling snacks and drinks.
Taxis are easily accessible and can be flagged down in the street, or you can call ahead to schedule a pick up.
Most locals ride bikes or walk. Just make sure you keep your bike locked.
Walk along the Paseo de los Turistas.