Jason,
My two cents worth: If you are visiting Florence for the first time, I'd stay there. The city deserves 3-4 days for a proper exploration. Lucca, at least the part within the walls, can be managed in a day or two. If the purpose of the trip is to see as much of the region as possible, I would stay in Lucca. It is cheaper than Florence, and there is plenty to see around there. Montecatini is a great place to spend a several hours, and if you're a Puccini fan, Torre del Lago Puccini west of Lucca is interesting, as well as the Puccini museum in Lucca. Florence is an easy, short train ride from Lucca. While in Lucca, be sure to try Osteria da Pasqualino. Reserve ahead of time.