Lucca is our home base. We have 13 days. We need guidance in deciding what road trips and towns to visit. At present we have 3 days Florence, and 1 day in Pisa to pick up family. We will have a car. So many choices: Siena, San Gimigano, Arezzo, Montecatini, Pistoia....
Here's a couple more...
Vinci, if Leonardo's inventions interest you plus the landscape is beautiful from the top of the Tower. Volterra, is less touristry than San G, and beautiful! Enjoy your trip!
I would do trips to places like Pisa and Florence near one another schedule-wise since you won't want your car for them anyway due to ZTLs. Couple that with the ease of train use from Lucca and to/from those places, it would be much simpler to dump the car altogether for a few days. What do you enjoy doing? There's a lot Lucca is convenient to. Places like Cinque Terre (another one to ditch the car for) would make great trips for hiking and outdoors-y stuff while the Chianti region, Montalcino and Montepulciano are great for touring wineries or quaint hilltop towns. San Gimigiano is a nice hill town to spend a few hours in, as is Volterra a bit farther down the road.
Brenda, Glad to see that you have included Pistoia, a real gem, and rarely visited. On any given day, you will be the only tourist around. The Piazza is only 9 blocks from the train station.
ANd, while in Lucca, be sure to rent bikes and ride the top of the rampart around the old city. Plus go to a Puccini concert some night. They are in a deconsecrated church and the one I went to was Puccini and Mozart and was awesome.
Viareggio on the coast is only a half hour or so away. It is a beach town but is also a major yacht building town. Also, an overnight trip to Cinque Terre, going through La Spezia is fun if you are looking for an alternative to hill towns. And along the way you can visit the marble towns Carrara and Pietrasanta. Lucca is a great base for exploring!