Taxi.
The city center is only 6km from the airport.
City imposed flat fee is Euro 25.30 between 10pm and 6am (during the day it's 22 euro).
There is also a surcharge of 1 Euro for each piece of luggage.
Whoever quoted you $60 is a scammer or a thief.
I've never spent more than 30 euro for a taxi to the airport, and my parents' house isn't even downtown but on the opposite side of Florence.
The two taxi cooperatives operating in Florence are below. You can call them at +39 055 4242 or +39 055 4390 to get a taxi or also use their apps.
https://4242.it/en/
http://www.4390.it/en/
All taxi drivers are independent operators and owners of their taxi cabs (licenses cannot be awarded to companies by law). However taxi operators are partners in taxi cooperatives to share certain services (like radio dispatch, marketing, etc.).