Florence has its own smaller airport called Amerigo Vespucci or simply Peretola, and it's located 4 km from the center of Florence. A taxi can make the trip in 15 minutes (15-20 euros), and the SITA/ATAF VolainBus ("Fly by Bus") between the airport and Santa Maria Novella (SMN) railway station takes 20 minutes and costs under 5 euros. If you are coming into Florence from many international destinations, you are probably using Galileo Galilei airport, in Pisa. This airport has a train connection from the airport sometimes requiring an immediate change in Pisa Centrale station to Florence (takes just over an hour) or a non-stop bus to SMN (takes 50 minutes). Pisa's airport has longer runways for slightly larger planes, but it is so much easier getting into Florence if you can fly direct to Peretola.
If traveling by train Florence's central station is called Santa Maria Novella. It is located in the northwest corner of central Florence. Trains going to Bologna, Milan, Venice, Rome and Naples depart almost every hour. Just remember to stamp your ticket BEFORE getting on the train. Many services, including two internet points, are located within the station. Be aware that there is a second station, east of the center (10 minutes by bus or taxi) called Campo di Marte, where some long-haul trains stop rather than pulling into SMN. The Trenitalia.it site has an English version, lets you research and buy e-tickets, and is generally quite informative.
If you're looking to use Florence as a base for travel to other Tuscany or Umbria hilltown destinations, often the best public transportation option is the extensive bus service. While visiting Florence you can either hop on a local bus or call a taxi. Taxis are expensive and cannot be hailed - you must call them to a destination or get one at one of the few taxi stands in the center of town. Tickets for the local bus services (ATAF) can be purchased at almost any newspaper stand. or tabaccheria.
There are bus routes all over the city, running frequently and cheaply, even up to Fiesole and Settignano, and you must purchase your ticket in advance (good indefinitely) ...you buy the ticket for 1.20 euro, stamp it when you get on the front of the bus (just like the train), it is good for 1 hour of travel...you can buy as many tickets in advance as you want, even 3-hour tickets and 1-day tickets, but remember to stamp it as you get on, in case an inspector checks en route. And there are plenty of regional bus services, to Siena, San Gimignano and so forth, with train service to more distant points.
