Florence is well-situated in terms of getting to other great destinations in Tuscany.  The Traveling Professor often does them on his small group tours to Italy.  Most are easy to get to from Florence's SMN train station or by bus.  Here are our favorites:

Pisa:  An hour train ride takes you to Pisa Centrale train station where the bus quickly takes you to the Campo dei Miracoli ("Field of Miracles") where the famed Leaning Tower of Pisa and Baptistry is located. If you can find a less frequent train to Pisa San Rossore stop, you can walk to the Leaning Tower.   Expect big crowds and lots of trinket shops, but it is definitely worth a visit. 

Lucca: A half hour by train from Pisa (1:20 from Florence) is the perfect other half day trip outside of Florence.  The walled city of Lucca is one of the best preserved medieval cities in all of Italy.  The Professor just loves renting a bike for a couple of hours to circle Lucca by riding on the wide bike path on top of the wall while peering into the gardens of expensive homes.  There is much more to see in Lucca like the botanical garden and the home of the famed Opera singer Giacomo Puccini. Climb atop Guinigi Tower to scour the landscape looking for invaders like they did hundreds of years ago.   Visit Pasticceria Taddeucci and try legendary dessert called buccellato made with bread dough, sugar, anise, anise liqueur, and raisins. Don't forget to have lunch (Lucca may have the best food in Tuscany) at a spot in the old Roman Amphitheater.  By the way, there is also a bus from the Field of Miracles directly to Lucca. 

San Gimignano and Wine Tasting at Tenuta Toriciano:  If going to Tuscany, a wine tasting is a must.  No one does it better than Tenuta Torciano.  Afterwards, about a mile away, visit the perfectly quaint and beautiful town of San Gimignano.  There is no direct train service, but a bus can get you to San Gimignano.  Contact Tenuta Toriciano about transportation to the winery.