There are regular buses around town, including from the train station to the Field of Miracles. Attractions are within a half hour walk of each other. Local bus tickets are available at tobacco shops; there are also vending machines both at the station and the airport.
Many car rental agencies are at the airport. While you will not need a car in the city itself, it can be a good choice if you want to go around Tuscany from Pisa. To get to the car rental offices take the shuttle bus in front of the airport's arrival hall to the right, close to the city bus stop.