Here's some things young teenagers might enjoy in Florence:
The Palazzo Davanzati (Museo della Casa Fiorentina Antica) is a 14th century townhouse you can tour to see rooms with furniture, painted “wallpaper,” kitchen tools, tapestries, bathrooms, interior courtyard, ceramics, lace, and other domestic objects.
Artist Clet Abraham cleverly alters the street signs in Florence and if you look, you can spot them while walking around. His workshop in San Niccolo is just off of Piazza San Niccolò. You can buy prints, stickers, postcards, etc. of his work. Throughout Florence there is also other street art that is fun to see.
See the view of the city from Piazzale Michelangelo or the even better and much less crowded view from the San Miniato church uphill. There are stairs up from the San Niccolo district or buses as far as the Piazzale.
The very readable and compelling “Diary of Florence in Flood” by Kathrine Taylor is an eyewitness account of the 1966 flood. There are photos and videos of the flood online, and on various walls in Florence there are markers showing how high the floodwaters reached at each location --- I believe the worst flood depth was 22 feet. Wikipedia says that the flood “killed 101 people and damaged or destroyed millions of masterpieces of art and rare books. It is considered the worst flood in the city's history since 1557. With the combined effort of Italian and foreign volunteers alike, or angeli del fango ("Mud Angels"), many of these fine works have been restored. New methods in conservation were devised and restoration laboratories established. However, even decades later, much work remains to be done.”
The Benozzo Gozzoli chapel in the Palazzo Medici Riccardi is a tiny jewel of a room where you could easily spend half an hour looking at all the details of clothing, animals, horses, flowers, etc. Binoculars are not a bad idea.
The lovely Cappella Tornabuoni in the church of Santa Maria Novella was painted by Ghirlandaio and his workshop in the late 15th century and shows many examples of Renaissance clothing and life. Again, binoculars.