In the town of Collodi, the birthplace of Carlo Lorenzini, the author of "The Adventures of Pinocchio", (between Florence and Lucca, next to Pescia), there is a Pinocchio open air museum and also a famous villa, or better a castle, called Villa Garzoni.
Carlo Lorenzini's father was a gardener at that villa while his mother was a maid, so Pinocchio's author basically grew up in it. In any case the villa has wonderful Italian gardens with water fountains, water falls, water games and even a labyrinth made with plants and bushes. I always enjoyed going there when I was a kid on some of our family Sunday getaways.
Also in nearby Montecatini Terme, a pretty spa town, I always enjoyed riding the funicular to the top of Montecatini Alto.
You can reach the above locations from Florence by train or bus. They are about 30-35 miles away from Florence.