I would pick the rental car up in Florence.
All rental agencies downtown are on Borgo Ognissanti. If you don't want to drive downtown pick Europcar rent a car. They have an office on 2 Via Forlanini, which would be close to the A11 freeway entrance to Lucca. From downtown it's 10 min by taxi. You could also pick up further out at the Florence airport at the rental center on via palagio degli Spini. However picking up there will incur airport fees and a higher taxi fare. Once in the car follow green sign directions to A11 to Pisa.
As you exit the Europcar garage on via Forlanini turn right in Via Forlanini. At the next light at the intersection with Viale Guidoni turn left. Then straight all the way. Once you pass the airport on your right, the A11 freeway starts. Lucca exit will be on the way to Pisa.
Once you are done with your Lucca stay, drive to La Spezia and return the car there. The La Spezia office for Europcar is near the port very close to the freeway exit. Take a taxi from there to the La Spezia Centrale train station. Hop on a train to Cinque Terre there.
I use a consolidator for rental cars:
www.kemwel.com
www.autoeurope.com
(both are part of the same corporate group and work the same way however prices differ at different times so check both websites)