To Florence from North America:
Florence can be reached via its own Peretola airport. Large airlines that fly into Florence from North America always stop somewhere else first:
Alitalia goes through Milan or Rome
Lufthansa through Munich (called Monaco in italian!) or Frankfurt
Air France through Paris
KLM through Amsterdam.

To Rome or Milan from North America: there are direct flights with Alitalia from Toronto, Chicago and New York.

Low cost options
Low cost airlines fly into Pisa, Bologna, Rome Ciampino, Ancona and Milan Orlio a Serio from the UK and the rest of continental Europe. Airlines include EasyJet, RyanAir, Meridiana. Flying to london, changing airports, and taking a low cost flight to italy is sometimes a way to save money though it's generally more tiring than worth its while.

From Pisa or Bologna airports to Florence you can take the train. The pisa-Fi trip takes 1.5 hours, runs hourly, and costs 5 euros on the regional train. No need to book in advance, just buy your ticket at the station. If the ticket window is closed at Pisa, there is no electronic ticket dispenser, but you can purchase it when you get to Pisa centrale (with most trains you have to change there).
The airports of Rome are connected with the main train station by train and metro.

Flights within italy
For long hauls within Italy, or to Sicily: meridiana flies within italy for reasonable rates; occasionally there are also specials.