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. You can rent bikes from any of the bike shops at really good rates. Just make sure you keep your bike locked. Helmets may be hard to find though, so you might want to consider bringing your own. Remember that pedestrians do not have right of way and that you must ride your bike in the street (not the sidewalk), so be careful!
Walk along the Paseo de los Turistas.