Lucignano is a very charming town - you can read more about it here:
http://www.discovertuscany.com/tuscany-destinations/lucignano.html
From the airport in Florence, you'll need to take the bus shuttle downtown to the SMN train station and from there head on south to Foiano della Chiana (passing through Arezzo). From Foiano, take a bus.
You can move around the area by bus. Expect some times to be long but you can definitely do it without a car, just be armed to plan carefully and have patience.
From Lucignano, you'll need to get to the towns on the train line - those are better connected. So, in the area, these are Castiglion Fiorentino and Foiano della Chiana, which you can get to by bus. From there, you can go to Arezzo. Once you're on the train line, you can then easily reach Florence back again.
Lots of buses leave from Foiano - to Sinalunga (to get to Siena) and Cortona, as an example.
You can get bus schedules here:
http://www.etruriamobilita.it/index.php?SEZ=16
and train schedules here:
www.trenitalia.com
Lucignano is in the "Valdichiana" - you can download the appropriate bus line map here:
http://www.etruriamobilita.it/index.php?SEZ=5
Siena and San Gimignano: you can reach Siena first by combination of bus and train.
San Gimignano is another bus from Siena (line 130, search for it here:
http://www.trainspa.it/train04/extraurbano.pdf )
Enjoy planning the trip!