What would be the best side trip from Florence, Cortona or Lucca or some other site?
You have many options: San Gimignano - 45 mins Siena 1 hour (take the bus not the train)
Lucca and Pisa... If you leave early, you can see Lucca and Pisa in one day.
Siena or Pisa are great day trips. Pisa is worth about 1/2 a day, but you can spend a full day in Siena wandering the medieval streets, the Duomo and the Piazza. Lucca is very pretty, easy to access, the train s you off just across the street from the walled city. To view the Leaning Tower, it's an easy 20 min walk to the Field of Miracles and as previous poster says, take the bus to Siena from Florence, train station is too far away.
Have been to the above-mentioned towns a number of times and we enjoyed all of them including Cortona but I personally think Cortona would be a bit far for a daytrip from Florence. Really enjoyed Lucca as it is much less touristy and less crowded than the other places mentioned. I know many people think Pisa is not worth a visit but we have been here 2x and our kids loved it. Climbed the tower both times and took the iconic photos of us holding up the tower which we love to look back on for a good laugh! Whatever location you choose to visit, you will have a great time. Buon Viaggio!