Transportation from Rome airport to Siena

Bus is probably your best option. I do not know pricing of the bus vs car service for you all, but I would recommend bus over train. The bus will take you to the center of Siena, whereas train will take you to the bottom of a large hill. With that many travelers of those ages, I think bus would be preferable.

I don't know if there is a bus that leaves directly from the airport. We had to leave from Tiburtina which is in the northeast quadrant of Rome, whereas the airport is southwest. Looking at the number of people and their ages, you might want to splurge on a car service just for piece of mind. You would be picked up at the airport and be in Siena in under three hours.

When we took the bus to Siena we first took the train from the airport to Tiburtina station. From there you cross the street and walk one block down to the bus station parking lot on the left side of the street. The bus to Siena (and buses to many other locations) stops there. We bought our bus tickets online at a discount but that locked us in to a specific time. We had factored in a couple of extra hours, being unfamiliar with Rome, but ended up getting to the bus station early and eating an early lunch at a restaurant across the street. There is a little office at the station where you can buy bus tickets. The bus leaves you in the center of Siena where you can easily walk to your hotel if it's in old town. If you look on Google maps you can see the buses parked in the lot and see the building where the offices of the various companies are. It helps in recognizing that you're going in the right direction.