Amerigo Vespucci Airport (FLR), formerly known as Peretola Airport and Florence Airport, handles mostly domestic flights and a very few international flights. Airfare tends to be somewhat expensive within Italy, so most people fly to one of the international airports in larger cities, and then take the train or drive to Florence.
Galileo Galilei Airport (PSA) located in Pisa, Italy may be your cheapest avenue to Florence. Busses and trains connect to the central train station from which you can take a train or bus to Florence's Santa Maria Novella Station. A number of low-cost carriers have service to Pisa from numerous European locations. Pisa is just 63 miles west of Florence, on the coast, and an enjoyable 80-minute train ride.