I agree, the best way to see the countryside with limited time is to rent your own car so that you can move around freely.
Florence is the best place to rent from, as you've already seen it then makes it easier to return the rental and then catch the train to Bologna.
There are no car agencies right at the train station, but they aren't far - most are located around Borgo Ognissanti which is just a few blocks south of the trains station heading toward the Arno river. You can see a map and more tips here:
In 3 days, I'd suggest staying in a central area in Chianti - anywhere between San Gimignano to Siena.
Here are some I'd suggest:
Depending on when you're coming (season of the year), some of the farmhouses will rent out apartments for less than a week, generally with a stay of 3 days. So if you do come in the low season, you should also look at
I'd suggest then doing Siena and a drive through Chianti on one day,
San Gimignano and the area nearby on another
and Val d'Orcia on another.
Pisa is further away so I'd suggest doing these on your 3 days - but if you really want to go, you could do a half day trip from Florence by train, prior to renting the car ;-). It is an hour train ride there and back, so you can definitely go in the morning, do lunch there and then return to Florence.
Hope you're enjoying planning the trip!