Two days in Florence is like trying to eat a gallon of gelato in two minutes.
I'm an architect so my biases determine what I see:
San Miniato al Monte, a beautiful Romanesque church away from the city. It's on a hill top and there is a great view of Florence. Santa Maria Novella, a beautiful facade designed by Leon Battista Alberti. ANYTHING BY MICHELANGELO-architecture, sculpture, and painting. A pivotal architectural design is the Laurentian Library (Biblioteca Mediceo-Laurenziana), especially the entrance lobby. This is part of the S. Lorenzo complex. Another design is the Medici Chapel also in S. Lorenzo. Don't forget his sculpture of David. The original is in the Galleria dell' Accademia, but you may choose to view a faithful reproduction in the Piazza della Signoria and avoid the Galleria. ANYTHING BY BRUNELLESCHI-the father of Renaissance architecture. He did the dome of the Florence Cathedral and most of S. Lorenzo, the Pazzi Chapel at Sta. Croce, S. Spirito, Ospedale degli Innocenti with cameo decorations by Andrea della Robbia (This was the first complete building of the Renaissance). The doors of the Baptistry of the Florence Cathedral. An incredible 3-D imagery of bas relief sculpturing. Palazzo Medici and other palaces; of city block sizes. The Uffizi-a painting and sculpture museum. Don't miss Botticelli's Birth of Venus and Michelangelo's paintings. People say the Sistine Chapel was not restored properly; that the restorations were not as Michelangelo would have painted. After seeing his paintings in the Uffizi, I think he did paint as the restorations show. After you are through with the Uffizi you will probably never want to see another Gothic or Renaissance painting or sculpture. Select carefully or you will waste a lot of time that can best be spent at another Florence attraction.
Except for San Miniato, almost everything is in walking distance. You won't be able to see all of the above in two days. And there's so much more.
More tips: If you want to buy good gelato check the banana flavor. If it is yellow, move on to another vendor who has a gray-colored banana flavor. This will assure a better quality of any flavor.
Watch out for women beggars. They usually have fake babies made of balls of cloth, but carried in their arms to look real. Many are also pickpockets. Do not let them close. There are also pairs of kids who operate as pickpockets. One approaches from the front and touches and pulls your clothing to distract you while the other approaches from the rear and steals things.
Another trick is to throw a piece of cardboard or newspaper in your face when you come out of a dark space into the light (or are otherwise disorientated.) The tendency is to reach up and remove the thing from your face, leaving pockets and purses exposed to their fingers. If you see the kids or anyone else heading for you, start yelling and motioning to them to stay away. I believe, Vada via means, go away.