Quick off the top of my head:
Lucca - Lots to do in Lucca itself.
Walk or bike along the Renaissance Wall (about 2.5 miles around). This can be a daily ritual. The top of the wall is a park.
Puccini Opera recitals (about an hour long) in Chiesa San Giovanni (a deconsecrated church) There is one every night that features arias from various operas by Puccini (born there) and other composers. Great acoustics.
Climbing the Torre Guinigi - a 14th Century tower in the center of Lucca. There are oak trees growing on top. 300 steps up but beautiful views.
https://www.visittuscany.com/en/attractions/the-guinigi-tower/
Lucca Botanical Garden - located inside the walls.
Villa Pfanner - Renaissance Villa inside the walls.
Several churches - Santa Caterina is one we like a lot - small but beautiful.
A few small historical museums.
Streets and Piazzas are nice to wander through.
A walk/hike from Lucca into the surrounding hills along a 19th Century aqueduct.
https://www.mykindofitaly.com/post/the-aqueduct-walk-lucca-to-vorno
Florence - Good day trip by train. Lot’s to see - The Uffizi Museum, Michaelangelo’s David, the Duomo (Cathedral), Pitti Palace and the Boboli Gardens, the Ponte Vecchio, etc., etc.
Pisa - easy train ride - can see the Leaning Tower complex. A good half day trip or longer if you want to explore the city itself.
Cinque Terre - Could do this in a day by train (or by car) to La Spezia and then a train to the 5 towns. Easy to get between the 5 towns, One good way to see a lot is to take the train from La Spezia to Riomaggiore (the southernmost town) and then take a boat along the shore to Vernazza. You will see the coastline. Then there is a good hike through the vineyards with coastal views to Monterosso al Mare (the northernmost town) and then train back to La Spezia (and train or car back to Lucca).
Pietrasanta- a town near the coast that is a an artist community city with outdoor sculptures throughout. Can get there by train.
The Garfagnana - The rugged mountain region just north of Lucca. It’s a car trip - there is a nice hike up to a fortress from Castelnuovo di Garfagnana.
https://www.montalfonso.it/en/fortezza/fortezza , A few interesting small towns en route, e.g. Barga and Bagna di Lucca.
Villa Reale di Marlia - A short bus ride (or car ride) from Lucca, Beautiful gardens and estate that was the former estate of Napolean’s Sister.
https://villarealedimarlia.it/en/about/the-history/
Carrera Marble Quarries - A car trip to Carerra to take a tour of the quarrying operations for Carrerra marble - the same marble that Michaelangelo and Bernini used for their classic sculptures - and the same marble that covers bathrooms and kitchens all over the world.
Montecarlo - A car ride into the wine country. Montecarlo itself is a cute small town, surrounded by vineyards and olive groves. Winery tours abound.
Montecatini Terme - A train ride to a town that has hot spring spas and an old part of the town that you take a funicular up to.
San Gimignano and Volterra - A longer car ride but easily doable. These are two Tuscan hill towns. Volterra has Roman ruins and San Gimignano is famous for its many 14th and 15th century towers.
Viareggio - A very short train trip to a beach town. You can walk along the beach, put your feet in the Mediterranean - or even swim. Forte di Marmi is another nearby beach resort town which is very upscale.
Giardino Garzoni - A train or car ride to Collodi (where the author of Pinocchio is from). there is a nice villa and gardens
https://www.visittuscany.com/en/attractions/villa-garzoni-and-its-garden/ Pinnocchio Land is in Collodi also if you will have small children in tow.
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