Any car rental website can find you a rental car for Rincon, Puerto Rico. Most likely you will pick up the car at SJU or BQN airports. If you want to try outside airports you can use Rincon's zip code of 00677. There is also a car rental agency, Angelos Car Rental, in Rincon if you just want to rent for a day or two, generally a bit more expensive than the airport guys, but convenient.
There are taxi services in Rincon, San Juan and Aguadilla. You need to arrange in advance. Once you are staying in Rincon there are several local taxi drivers who can take you around with less notice. But keep in mind, Rincon does not have taxis driving around looking for people who need a lift. A Rincon taxi service will drive you local or on a sightseeing tour. Rincon Taxi 787-398-3621
You can walk in Rincon but many areas have steep hills, best to know the terrain where you are staying.
You can rent bikes to ride in Rincon, but again, some areas have very steep hills so make sure you know the terrain near your rental.