For a very authentic dining experience with a local crowd, consider Casprini da Omero in Passo dei Pecori. This very simple family owned and managed place is easily reached on the bus that runs between Florence and Greve.

Momma is in the kitchen every morning making fresh pasta. Produce is grown locally. The Bistecca Fiorentina is to die for. Unlabeled house Chianti, probably from the adjoining Nozzole estate, goes for 6 euros a bottle.

Menu and more info here:
http://www.cdaomero.com/