The OP provides little indication of what's really important in answering questons about a place like Florence (unless you're on a guided tour), which is background. My wife and I are fortunate in that (even though Americans) we are Catholic and old enough to have had a traditional education, and we both had an interest in history and art that predated any foreign trips. This always strikes us when we see tourists walking through something like the Accademia or Bargello in Florence or Mont-Saint-Michel in Normandy and realize from the glazed expressions that they have no real idea what they're looking at.

So any effective answer must depend on background. That said, Accademia, Uffizi, Museo del Duomo, Bargello, Duomo, Baptistry, Santa Maria Novella, and a personal favorite of mine- the Brancacci Chapel in Santa Maria del Carmine, where you can see in Masaccio's frescos the birth of Renaissance painting. And more. And don't forget some limoncello after dinner.