Pisa, Florence, Lucca
Ciao rel3,
Since this will be your first time in Italy, I wholeheartedly recommend you see Pisa and the Leaning Tower. So your first night has to be in Pisa - the airport isn't far from the center and a taxi ride costs from 6-8 euro on average to the center.
You can spend the morning in
Pisa, have a delicious lunch at
Montino's pizzeria (get a good size portion of
cecina, a local speciality) then head to Florence by train for the next two nights. Since June 24 is the feast day for Florence's patron saint, St. John the Baptist, the city will be celebrating with a fireworks display at night, enjoyed by everyone from along the Arno river. You can just enjoy outdoors Florence on this day, spending the next day visiting any churches or museums you wish to see from the inside. As you already know, with a young child it is best to remain flexible and do things (or not do) as you're there.
I recommend a visit to the Boboli gardens, even if there is no playground, the beautiful grassy area, fountains and shaded areas are perfect for spending time outdoors with a toddler. Seek a spot where you can spread out a blanket and let the little one play around.
On your last day, you can take the train to Lucca and spend the day there, it is a quiet Renaissance town, completely walled. You can rent bikes with a child basket and ride all around the town on top of the walls, very popular with runners, cyclists and just anyone wanting a stroll and good view of Lucca. From there, take the train to Pisa to catch your flight back home.
The only problem might be what to do with your luggage while you visit Lucca as I've heard there are no lockers or baggage deposit at the station. I'll have to look into this and will let you know if I find something more!