Florence’s bus company is called ATAF, and it operates in the city centre and to an extensive area further afield, like up to Fiesole in the hills (number 7 bus from the train station), Scandicci and Campi Bisenzio.
Buying bus tickets are easy: you can either pick one up at a tabaccheria or buy one onboard the bus. ATAF also has an option to purchase a ticket via text message to the number 4880105, with the word “ATAF”. You must wait for the confirmation, which you can use as your ticket. This service is only available for TIM, Vodafone, Wind and 3 Italia. Tickets purchased ahead of time cost 1.50 euro, onboard 2.50 euro and via text message 1.80 euro. They all last for 90 minutes.
For a full list of routes and timetables, check out the website.