The Uffizi Gallery: This is the big ticket, one of the world's greatest museums. For the love of Botticelli, Titian and Tintoretto, make a reservation for entrance so you don't have to stand in multi-hour lines, especially during the peak summer holiday period.  Make a reservation for a timed ticket via phone (phone numbers are on http://www.uffizi.firenze.it/) or book online through an agency such as http://www.weekendafirenze.com.  In the winter, its possible to walk to the front of the museum entrance, pay for ticket and enter immediately, if you arrive at opening time. 

The Galleria dell'Accademia:  This smaller gallery won't take as long as the Uffizi, but go and say hello to Michelangelo's David in the spectacular flesh, er, marble. Don't forget the very nice rear view.

The Bargello: Don't overlook this breathtaking collection of Tuscan Renaissance sculpture, including Donatello's David.

La Specola-Museo Zoologico, the city's natural history museum. Hurry through the fossils and mangy wildlife displays to the rooms that display the museum's collection of 18th century wax human anatomical models. These are what doctors once used to learn about the human body (when there was nobody around to dissect). Some - including dioramas of plague victims - depict unfortunate medical conditions; other show various aspects of healthy bodies. They are grotesque, masterful, informative and wonderful. The museum is on the other side of the Arno (the Oltrarno) just past the palaces.