Airports

Florence Airport (FLR) is four kilometers from the center of town. It's too small to have direct flights to and from the U.S., but there are direct flights to and from Amsterdam (KLM and Vueling), Barcelona (Vueling), Brussels (Brussels Airlines), Frankfurt (Lufthansa), Geneva (Swiss Airlines), Lisbon (TAP), London (British Airways and Vueling), Madrid (Iberia and Vueling), Paris (Air France and Vueling), and Rome (Alitalia), among others.

Pisa International Airport (PSA) is about an hour away from Florence, but is very easy to access via airport transfer or train. It's much bigger than Florence Airport, and often has less expensive flights. There are direct flights to/from Athens (Aegean Airlines), Barcelona (Vueling), Copenhagen (Scandinavian Airlines), Dublin (Aer Lingus), Istanbul (Turkish Airlines), London (British Airways), Munich (Lufthansa), Prague (Czech Airlines), and Rome (Alitalia), among others.

Booking Tips

Instead of searching for round-trip tickets between the U.S. and Florence/Pisa, try running separate searches for (1) flights between the U.S. and one of the "connection" cities above, and (2) flights between that connection city and Florence/Pisa. We did this recently; it made a $500 difference per traveler.
Use the "explore" function on Google Flights to identify the least expensive "connection" city for purposes of your hometown.
Use the hopper app to set up fare alerts, and determine the least expensive time to buy.
Consider using FlightFox --- the awesome travel concierge that we used to book our own flights for the wedding and honeymoon. There's a fee, but they won't charge it unless their fare quote saves you more than the amount of their fee. They specialize in maximizing frequent flyer miles and credit card points.