Car rental

Train to Chiusi and rent there.

Could also just rent it upon arrival in Siena.
There are rental car concessions a short walk or taxi ride from the train station, and cheap underground parking in front of the station itself, with easy access via a series of escalators up into the heart of the town. A taxi will cost less than 15 Euro, and is probably worth it for the door to door service with luggage.
From the underground lot it's an easy exit onto the main highway out of town to explore either Chianti to the north or the Val D'Orcia to the south ... well away from the dreaded ZTL's that cause so much angst for first time drivers in Italy.

Edited to add: I see that train connections between Rome and Siena are pretty convoluted - taking longer to get to Siena than I thought at any rate - so collecting your rental car on the outskirts Rome (you really don't want to drive in the city) and driving to Siena might be your better option. Chiusi would work as well.
Many here (us too) rely on AutoEurope - they offer competitive pricing and a clear explanation of the insurance options. Suggest paying a little extra for the zero deductible coverage - it's worth it for the peace of mind. Ensure that you get an automatic too. Booking a car with manual transmission would be a lot less fun than you think.

We have picked up cars in Rome and at FCO. The airport location certainly is easier and you get on the Autostrada easily.

What to do in this case is dependent on whether or not you wish to stop at any location south of Sienna. We have driven out of Rome to Chiusi and made a stop in Civita di Bagnoregio for lunch. It would be possible to do the same if you wished to drive to Sienna in one day. On our trip we avoided the toll roads, which makes for a longer trip, but can be more scenic.
We used San Gimignano as a base rather than Sienna for touring the region. In fact, almost any town in the area could be used as a base; it does not need to be a major town. Regardless, if you stay near or in Sienna, parking will be a challenge, but it is in any city. We have accepted that as the price of using a car.
The big advantage of staying in a city like Sienna is experiencing it in the evening when the tourists have left. The character of the town changes and becomes much more enjoyable. When we stayed in San Gimignano, when we went out in the evening to eat, the town was quiet and much more intimate. Early in the morning, before the tour buses arrived, we also could enjoy the sites.
Sienna is a city to be visited for its history and buildings.
We did not find it difficult to drive out of Rome, if that is what you end up doing. We rented a car near the north edge of town and drove out of town without any problem.
We used Gemut, a USA company to arrange for a car. We found them helpful and they did a lot of the leg work for us.
Enjoy your trip.

It is not that hard to get to the Autostrada from Roma Termini rental car locations, and easier still from Roma Tiburtina, which is right at the inner ring road. No need to go to the airport, which is in the wrong direction.

Another vote for picking up at Tiburtina. Very easy to get on the Autostrada, and you’re on the right side of Rome to head to Tuscany.

The most popular place north of Rome for a rental car is Hertz in Orvieto. . It's a quick 70 minute ride from Rome Termini. Best thing is that the Hertz office is across the parking lot at the train station.

There's a controlled access highway north and then a 4 lane highway right into Siena--a 90 minute drive. (We were traveling south, and turned our car in at Orvieto.)

Warning: Siena is a very congested city for automobiles. We stayed outside of the city--west of Poggibonsi and San Gimignano. Virtually every farm in Tuscany has rooms or apartments for rent as a source of secondary income.

"Siena is a very congested city for automobiles"

Actually, every Italian city and town is congested, which is why you'll need to familiarize yourself with local driving restrictions before getting behind the wheel in Italy - particularly those dealing with the ZTL's that exist in virtually every hill town in Tuscany and Umbria. Inadvertently venturing into one will result in a stiff fine. A little of advance homework will probably keep you out of trouble.
We thought that Siena was one of the easiest towns to get in and out of, particularly if you park at the underground lot fronting the train station that I mentioned previously. It's right on the SR2 route that will take you safely out of town in either direction - you never have to go anywhere near the restricted ZTL zone.

Has anyone ever used Fox car rental at Malpensa?