I don’t recommend flying in to the Florence airport in general—it’s cheap and convenient to get to Pisa or Rome by train—but it may be the case that your itinerary just works out that way. There is an “official” airport shuttle bus which will take you to the Santa Maria Novella train station for 4.50 Euros (about US $7). But if you are intrepid and very, very cheap, you can take a regular city bus to and from the airport in about the same amount of time for just 1.20 Euros. This is an especially reasonable option if you have to get to the airport from downtown in the middle of the day.
I’ll give the instructions from the train station to the airport. To get to the train station, reverse them.
You need to take the 29 or 30 bus, which go every 20 or 30 minutes through most of the day.
First, buy a bus ticket. There are several newstands that sell bus tickets inside the train station.
The buses start on Via Luigi Alamanni, near the NW corner of the train station. If you are in the train station, facing the trains, turn left, walk out of the station, cross the street, turn right, walk down the sidewalk and cross one street. The bus stop (along with many others) is along this block. The train station is ringed with various bus terminals, so you’ll have to be careful to find the right one (29 or 30, they’re at the same stop).
Get on the bus and, of course, stamp your ticket. The trip takes 15 or 20 minutes. The bus will take a few turns and then travel primarily down Via Francesco Baracca, a big wide thoroughfare with strip malls and gas stations on it. You want to get off on Via Pratese. You’ll know you’re getting near it when you pass a big fork in the road (Via Pistoiese) and you’ll know you’re there when the bus lurches decidedly leftward after having travelled straight a good long while. Press the button to get the bus to stop.
You’ll be on the North side of Via Pratese. Here’s a map that might help. Walk back the way the bus came to Via Del Motrone and turn left. Don’t fret, you’re almost there. The street winds for 100m or so and you’ll reach a big complicated intersection with an overpass. That’s the A11 freeway. If you peer under the overpass, you’ll see the entrance to the airport.
Cross this intersection, being careful not to die. You are now at the airport. Congratulations, you have saved yourself 3 Euros!