jen,
For driving in Italy, note that each driver listed on the rental form will require the compulsory International Driver's Permit, which is used in conjunction with your home D.L. These are easily obtained at any AAA office and are valid for one year. Failure to produce an IDP if requested can result in hefty fines, which will be collected on the spot. As someone else mentioned the Zona Traffico Limitato (ZTL) areas will also be an issue (more hefty fines).
As others have mentioned, using a rental car to tour around that area is not the best idea (IMO), especially as you have a limited time in port. You'd also have to allow for the time to get to the rental agency, do the paperwork and then the time to return the vehicle and get back to the ship. There's also the possibility of getting lost for a short time, which will reduce your time in the places you want to see. The five towns of the Cinque Terre are especially not a good place to tour by car, as parking could be an issue at that time of year, except in Monterosso as Laura mentioned. Using the local trains is a quicker and more efficient method to see the five towns. Travel time from La Spezia Centrale to Riomaggore (the closest town) is only about 9 minutes. You'll waste far more time than that if you drive.
Train travel time from La Spezia Centrale to Pisa Centrale ranges from about 38M to 1H:27M, depending on which train you use. The trains in Italy are very easy to use, but there are a few "rules" that you need to be aware of. If the train you choose is a Regionale, you MUST validate the tickets prior to boarding the train or risk hefty fines, which will be collected on the spot. Tickets for the Intercity and Freccia trains do not have to be validated, but those come with a compulsory seat reservation that's specific to train, date and departure time, and can only be used on the one train listed on the ticket.