Visit a Secret Bakery
These are not your typical bakeries that are open during the day. Instead they operate a night, usually beginning around midnight. Pull out your map of Florence, or better yet your smartphone! Then use this link to find a secret bakery in Florence. Secret bakeries can be a little tough to locate, but it is worth the effort. Make sure you go in a group, follow your nose and whisper when you get close. Once you find it, politely knock and exchange a few euros for a bag of delicious Nutella filled pastries.
Spend the Afternoon or Evening at Piazza Santo Spirito
The Santo Spirito Neighborhood was my temporary home base when I studied abroad in Florence. This area on the other side of the river is a great place to escape to after a morning in the city center. Unlike most popular areas in the city, this neighborhood isn't overrun by tourists. Instead, you'll find a neighborhood full of artisans and locals. Once you reach the neighborhood, grab a pizza from Gusta Pizza, sit on the steps of the Basilica di Santo Spirito and enjoy the ambiance as you devour your box of cheesy deliciousness. If you visit during the day seek out some of the small artisan shops in the area where you can buy beautiful handmade goods.
Bring your camera! This location offers am amazing view of Florence from up high. Most people visit during the day or in the evening. It can get crowded with tourists trying to snap the perfect picture of the city and Il Duomo. If you're up to an early morning, I'd suggest visiting to watch the sunrise. Grab coffee or breakfast from one of the kiosks at the top. One word of advice, if you aren't up for the long hike to the top, take a taxi or catch a bus.
If you happen to be wandering the streets of Florence near Il Duomo at night, be sure the head down the street and stop by Eby's Bar. At Eby's Bar, Eby himself will make you some of the most beautiful shots that contain some of the most unusual ingredients: Nutella, green apple liqueur, a little hot sauce. They are true works of art. As you take the shot, Eby and the staff will yell at you and you will like it. The only way to truly understand the experience is to visit Eby's Bar yourself.
As part of a visit to Basilica di San Lorenzo, you should stop by the Laurentian Library. The staircase and reading room of the library were designed by Michelangelo. Commissioned in the 1520's by the Medici family, the reading room was originally meant to house the family's large collection of books. Due to the library's somewhat strange location, it is often overlooked by tourists, but is relatively easy to find.
Honorable mention: I can't complete this post without mentioning the Palazzo Vecchio. Full of ornate and hidden features Palazzo Vecchio is most likely on your list. However, if you can catch a tour it is worth it because there are so many details you would otherwise miss. For example, In one of the rooms there is a vent - perfect for viewing and eavesdropping - overlooking the Salon di Cinquecento that is cleverly hidden by a painting in the great room.
Read more about my travels in Italy here, here and here.