If traveling by train Florence's central station is called Santa Maria Novella (Search for Firenze SMN if you are booking online). 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 (www.Italiarail.com), 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 hill town 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, or often at your hotel.There are also 4-way tickets with a broad arrow top and bottom on both sides, good for 4 separate uses. 

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 90 mins (previously 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.

As most of the streets are one way, the bus stop for your return journey may be located as some distance. But it should not be a problem as you can always take the bus going further ahead & continue your return journey after it loops at the terminal.(& enjoy the beautiful city enroute)

And there are plenty of regional bus services, to Siena, San Gimignano and so forth, with train service to more distant points. 

To emphasize the importance of Validating your ticket on transportation, the fine if you are caught failing to do so is a heft 200 times the value of the ticket, which for a Eur 1.20 bus ticket is Eur 240 or $300! 

For buses to Siena the line used is SITA whose terminus is to the east of Novella train station, every hour except during lunchtime. Buses to Lucca are with LAZZI line whose ticket office is round the corner from MacDonal's facing Novella. Their buses leave from around the next corner.