I’m glad to help you plan your two weeks stay in Tuscany 🙂
While Tuscany can be best explored by car, you can still enjoy it also taking public transport, only need to plan it well!
I second your decision to stay in Lucca for 3 days. Lucca is an extremely charming town, within the walls you have anything you can desire and the train station is right next the huge walls, so that’s very convenient to go for day trips or change base.
Lucca itself deserves some of your time, while near Lucca you find also interesting seaside destinations like Viareggio, lively town on the northern Tuscan coast offering long beaches of sand, lots of restaurants, and all kinds of services. You might well stay in Viareggio for a few days in case you’d like to enjoy some beach life. Viareggio is only 20 minutes away by train from Lucca.
Among the many day trips you can make from Lucca, you might want to visit Pisa, famous for its Leaning Tower, Livorno, nice town on the coast, and Montecatini Terme, renowned for its thermal baths.
After Lucca and Viareggio, I suggest you reach Florence and make it your base for most of your stay. Florence is the town with most connections in terms of public transport. That way you can fully enjoy Florence, that really has a lot to offer, and use train and bus, located in the city center, to reach some of the most famous and popular destinations in the region. Here are my suggested popular day trips from Florence and here are other side trips to enjoy.
In case you wish to add a further base to your holiday, then I’d suggest the beautiful Siena, where you’ll find a lot to see and do while taking advantage of bus service to reach the wonderful hill towns in the area, like Montalcino, Pienza, or Montepulciano. Take a look at my suggested trips from Siena.
Feel free to get back for any other question,
Ciao from a sunny Florence!