Bus Transportation from Florence Airport to Greve

Train station is not at the airport
You will need to take the tram into the center

Bus from Greve to Siena or Montepulciano is very time consuming

This is the bus company
https://www.at-bus.it/en

T2 tram line links the airport to the railway station. It takes about 20 minutes so walking is out of question.

I am afraid buses to Greve have been moved from the standard bus station to a new bus hub in piazzale Montelungo, on a side of Fortezza da basso, between the Fortezza and the railway tracks. The fastest way to get there would be a taxi, the second fastest T2 tram to Valfonda stop, enter the railway station and walk along track 16; at the end of track 16 there is a metal bridge and a walking ramp descending to Montelungo. Walking time is 5-6 minutes.

Buses to/from Greve are local buses geared to commuters, no direct service to Siena, and you probably have to go back to Florence to get to Siena. Probably you do not have to go back all the way at the railway station as you could interchange buses at Porta Romana, but I am afraid this is not too clear and easy for a tourist.

Getting to Montepulciano on public transportation from Florence, let alone from Greve, is long and difficult.

You'll need to take the bus from Florence to Greve which probably means bus 365A.
The Tuscan bus planner is here: https://www.at-bus.it/en/lines-and-timetables

I find Italian bus company websites frustratingly unuser friendly and unnecessarily complicated but the stop you'll probably want in Florence is: FIRENZE HUB Montelungo which is on the right side of the train station (facing it) about one long block up. These bus stops are well marked and it should be easy to figure out which one you want. This isn't always the case - especially in small towns - so pay attention to where the bus lets you out in Greve so you know where to come back to. As small as Greve is there are 3 stops - essentially before town, in town and just after town so check on where you want to get off.

You should take a taxi or the tram from the airport to the train station and then switch to the bus. The taxi will be much more expensive but will drop you right at the bus stop. The tram #2 stops on the west side of the train station at Alamanni - Stazione Santa Maria Novella but then continues around the station to Valfonda - Stazione Santa Maria Novella stop on the other side. From here it is probably a 6 minute walk up the side of the train station to FIRENZE HUB Montelungo.

It also looks like both the tram and the bus converge on the next stop Fortezza for the tram and BESLAN T1 FORTEZZA for the bus which are right across from each other. Note: This is just from reading the tram and bus schedules I haven't actually done this transition but it seems like the most direct solution without walking with luggage and you avoid the always busy SMN train station.

The Fortezza stops show in the Piazza - hopefully this link works for you.
https://www.google.com/maps/dir/Fortezza,+50129+Firenze+FI,+Italy/Beslan+T1+Fortezza,+50123+Firenze+FI,+Italy/@43.7804909,11.249106,292m/data=!3m2!1e3!4b1!4m14!4m13!1m5!1m1!1s0x132a56a7027e0a11:0xe968d6433f1bb4d3!2m2!1d11.2496611!2d43.780511!1m5!1m1!1s0x132a56a6fd8b3ae7:0x1b4bc2d075fe2985!2m2!1d11.2496103!2d43.7802403!3e2?entry=ttu&g_ep=EgoyMDI1MDMxMi4wIKXMDSoASAFQAw%3D%3D

As others have said connections between Greve and Siena are difficult. It's probably less than an hour drive but reliable connections probably mean heading back to Florence as the local hub and then going to Siena. There may be some Greve to Radda or Castellina bus connection and then onto Siena but that would probably be once or twice a day not a reliable connection.

Hope that helps,
=Tod

If you are not willing to rent a car then you should stay in either Florence or Siena, not Greve.

Buses are not frequent and mainly concentrate at peak hours to serve commuters and students. They are almost absent on Sundays. From Greve to Siena you need to go back to Florence then down again to Siena, a 2+ hour journey. From Siena to Montepulciano it’s another 1.5 hours from Siena. I doubt you can do the trip back and forth on the same day.

Simply put you need a car in Greve unless you are content with spending most of your time in Greve
It just doesn’t have good public transportation

An alternative to car rental is to hire a prvateydriver for a day ir 2