Ciao S!
Welcome on the DT Forum! Glad to hear that you are planning a memorable family trip to Italy.
Tuscany offers plenty of solutions for the families with kids. Considering that touring with kids has a complite different pace I would suggest you a location that only by itself is a guarantee of comfort and kid-friendly activities.
One property which is extremly close to Florence offers orchards, kitchen garden , pet garden, and elegant apartments for rent. The property is
Villa il Paradisino and is located in Sesto Fiorentino, at the edge of Florence.
I feel like suggesting it because you can exactly find playgrounds withing five minutes on foot, green meadows, Medici Renaissance villas, lots of Italian families playing outdoor in a quiet neighbourhood where everybody walks or ryde bikes instead of taking cars.
There are tracks along the torrent (20cm of water, so, not dangerous at all) walking on small medieval bridges, Costanza (the hoist at the villa) bakes with kids and families fresh pasta and pizza in her 20mq kitchen !!!
They have an extraordinary property where you can breath history, enjoy silence, herbs, pick veggies and fruits anytime.
The Villa is not far from the local train station: by train the connection to Florence is only 12 minutes!
So, a perfect location if you wish to visit the town with a kidfriendly guide, have gelatos, climb a medieval tower and do fun activities for the entire family.
From Florence you can easily plan a full day by train to Pisa and from Sesto there is a direct connection to Lucca.
I do think it would be the perfect stay to accomplish many necessities, both for parents and kids!
Good food, relaxing spaces, playgrounds, kidfriendly museum activities.
I am Florentine and I decided to settle in Sesto Fiorentino exactly because of the peacefulness and close distance to downtown.
You'll love it! And by the way, I have a 4year old daughter, we will be glad to show you the closest parks to play together! See you in Sesto!

Elena