It's possible, but it's better to take the bus because it drops in the center.
http://www.sitabus.it/sita-toscana/Firenze-Siena06.pdf

You can reach quite a few towns by train/buses. Just depends on where you want to go.

We have been looking into this as well. The train station is at the bottom of the hill in Siena and you would either have to walk or take a bus to the top. The bus, as kybourbon mentioned, drops you off in the center and my understanding is easier to maneuver. There are two different bus services, however, one is a local and the other is direct.

the bus station in florence is about 100 metres away from the main SMN railway station, to your left as you face the front of the station.

this is the website for the buses - timetables for Firenze [florence] to Siena and San Gimignano are shown in the middle panel.

http://www.sitabus.it/wps/portal/DovePartire_Toscana

you can also get a local bus up to Fiesole, about 30 mins journey north of Florence from the centre or from the piazza san marco at the top of via Cavor [round the corner from the Accademia]. a good idea is to visit the monastery in piazza san marco first to see [mainly] the frescos of Fra Angelico [it's only open in the mornings] then get the bus up to Fiesole. the monastery of san Francesco is quite interesting [fascinating free museum in the basement and make sure to see the nativity scene in the church] and the amphitheatre is also good to wander round for an hour or so; the restaurant in the central square was very good, and there is also the Basilica to see.

Bologna is easily reached by train from Florence; it takes about 30 minutes.

that should keep you busy for a week or so!

There are regular hourly or so trains from Florence to Pisa, Siena, Lucca and other locations.

No reservation necessary, just turn up, buy a ticket from the touch-screen self-service machines with English language faciity, remember to validate your ticket in the yellow platform machines, then hop on the next train.

Florence-Siena is about 8 euros each way - www.trenitalia.com will confirm train times and fares IF you pick any date in the next 7 days (it doesn't show regional train fares more than 7 days ahead, I have no idea why!)

Yes, it's a bit of a trek up the hill at Siena, but I've done it (I hate buses if there's a nice comfy train instead!). And Siena is worth the walk!

Thank you all so much. This is why I love this forum.

While there are regularly hourly trains to Siena, there are also regular hourly buses that get there faster than the train (15 minutes faster). Training will take you at least 45 minutes longer by the time you add on getting to the center of Siena. The buses Florence/Siena will have AC and the train might not as it's the older R trains (nothing comfy about them) on this route. It's also more than just a little walk up a hill (Mappy says it's a 27 minute walk). The buses are not like city buses, but more like long distance tour-type buses. It is possible to catch a city bus from the train station. You have to go in the underground parking lot across the street (search for stairs or ramp) to catch it (buy a ticket in the gift shop in the train station or in the ticket machine underground).

I would be tempted to split the Florence time and stay 3 days in Siena. Siena has bus service to Montalcino, Pienza, etc. This is a map of the bus lines/numbers from/to Siena.
http://www.sienamobilita.it/mappe/Mappa_EXT.pdf

The Florence tourist office was offering a Chianti tour on Thursdays for 15€ which included a wine tasting, but it might be seasonal.
http://www.firenzeturismo.it/

Any bus/train you take, make sure of the return times so you don't get stuck somewhere. Some may not run very late.

If you zoom in on page 1 and 2, the cities surrounding Florence are listed along with the transport options to get there (name of bus company or bus number/train, etc.). Zoom in on page 2 and the map shows you where the various bus companies are located around the train station.

http://www.firenzeturismo.it/images/stories/informazioni-turistiche/come-muoversi/in_bus/oltre-firenze.pdf

Thank you, Kybourbon. You are a wealth of knowledge. I will consider your advice about splitting the time between Florence and Siena. Thank you for the links.