There are many museums in Florence worth it to visit beyond the Uffizi Gallery and the Accademia Gallery. The Museums in Florence are divided into 3 groups: 

State museums run by the Italian national government
Florentine Civic Museums run by the city
Private Museums

REMEMBER THAT ALL OF THE STATE MUSEUMS offer FREE ENTRANCE the FIRST SUNDAY of every month!!

But which are these museums? 
This is a list that italianme has prepared for you, so you can enjoy your experience through the Florence Museums in the best possible way:

STATE MUSEUMS

Uffizi Gallery (now called the Uffizi Galleries, which also includes the Pitti Palace and Boboli Gardens. The Contini Bonacossi Collection has now been integrated within the Uffizi Gallery collection)
Accademia Gallery (includes the Cherubini Conservatory Musical Instruments Collection)
Bargello Museum
Medici Chapels
San Marco Museum
Pitti Palace that holds in its interior:
– The Palatine Gallery & Royal Apartments
– The Gallery of Modern Art
– The Museum of Costume and Fashion
– The “Treasury of the Grand Ducks” called also: Medici Treasury or Silver Museum
– The Porcelain Museum
– The Boboli Gardens

Palazzo Davanzati Museum
Casa Martelli Museum
Church and Museum of Orsanmichele – free entrance
Perugino Hall – free entrance

THE VILLAS – museums also run by the Italian State, but with a changing schedule that applies only to them. They are also called the “Medici Villas”:
Garden of the Medici Villa of Castello – it is not possible to visit the inside of the villa
Corsini Villa of Castello
Medici Villa La Petraia
Medici Villa of Poggio a Caiano and Museum of Still Life
Medici Villa of Cerreto Guidi and its Historical Hunting and Territorial Museum
The Last Suppers and Cloisters

Other smaller State-run museums in Florence with free entrance are:
Cenacolo of Ognissanti – Last Supper of Ognissanti
Cenacolo of Andrea del Sarto – Last Supper of San Salvi
Cenacolo of Fuligno
Cenacolo of Sant’Apollonia – Last Supper of Sant’Apollonia
Cloister of the Scalzo

Temporarily closed museums:
The Carriage Museum (Pitti Palace)
The Vasari Corridor (Uffizi Gallery) – closed for restoration until further notice.

FLORENTINE CIVIC MUSEUMS

Palazzo Vecchio Museum and Tower in Piazza della Signoria
Brancacci Chapel in Piazza del Carmine
Santa Maria Novella church and cloisters in Piazza Santa Maria Novella
Stefano Bardini Museum in Via dei Renai, 37 (Ponte alle Grazie)
Novecento Museum in Spedale delle Leopoldine P. Santa Maria Novella 10
Salvatore Romano Foundation in Piazza Santo Spirito, 29
San Niccolò Tower in Piazza Poggi
Forte di Belvedere in Via San Leonardo, 1

PRIVATE MUSEUMS

Palazzo Strozzi in Piazza degli Strozzi
Horne Museumm in Via dè Benci, 6
Stibbert Museum in Via Federico Stibbert, 26
Fondazione Casa Buonarroti in Via Ghibellina, 70
Museo Salvatore Ferragamo in Palazzo Spini Feroni, Piazza di Santa Trinita, 5/R
Museo Stefano Bardini in Via dei Renai, 37