We are staying at a villa in Greve for a week, but would prefer to take a train to and from Florence rather than a bus. Can anyone tell me where the nearest train station to Greve would be that has service to and from Florence?

Greve is all of 20 miles from Florence. I suppose you could navigate (by bus?) the 15 miles to Figline Valdarno, then take one every hour 30-45min local milk wagon train into Florence.

But why not just take your rental car? Of if you're spending the week at a villa without a car, then take the bus. Or if you're going to take a taxi 15mi to the RR, then take it 20mi into Florence.

There is no feasible train.

A train will take you way out of the way and take a long time. Your rental car will create parking issues. The bus from Greve to Florence is very efficient - it takes less than a half hour and drops you right next to the train station, very convenient to the sights you will want to visit in Florence. The bus runs frequently.