There are hundreds of books in English about the history of Florence, quite a number of them centered on the medieval time period. If you read Italian, your choices multiply. I won’t attempt to create anything like a full bibliography here, but if you are new to this time and place and would like to find out more, the following list would be a good starting point.
I’m including three modern history books, and two contemporary chronicles. The latter are available, in English, online, or can be purchased in print versions.
For a good all-around history of Florence at this time, I would recommend A History of Florence 1200-1575, by John M. Najemy (2008). This is a very readable, succinct work by a noted Florentine historian.
To learn more about the magnate families and how they fit into the fabric of Florentine society, I would suggest The Florentine Magnates: Lineage and Faction in a Medieval Commune, by Carol Lansing (1991). This book is bursting with fascinating anecdotes, and it also manages to put the phenomenon of magnate society into a context and trace its development and decline.
To get a sense of how the church affected the lives of all medieval Florentines, and of their beliefs and rituals and practices, I recommend Florence and Its Church in the Age of Dante, by George W. Dameron (2005). Everything you might want to know about the church in medieval Florence is here.
Florentine chronicler Giovanni Villani lived from about 1276 to 1348. He began writing his Nuova Cronica after traveling to Rome for the Jubilee celebration in 1300. When he died in 1348, probably of the plague, his brother and his nephew continued his work. You can find a free Kindle version of the Cronica (selections only) translated by Rose E. Selfe and edited by Philip H. Wicksteed, or follow this link.
Dino Compagni, a Florentine contemporary with Dante, was both chronicler and political figure. Born around 1255, he lived until 1324, and his Cronica spans the period 1280-1312. Selections are available translated by Daniel E. Bornstein in Dino Compagni’s Chronicle of Florence (The Middle Ages Series) (1986). Alternatively, for an earlier translation, follow this link.