car rental and trains
I also highly recommend using the train between these three large cities (Rome, Florence and Venice) as they are well connected with fast trains and are very convenient to use. You don't need a car to visit them, as walking is the best way to see both Florence and Venice (there aside from walking, your only alternative are water taxis, buses and gondolas

) and in Rome you can walk, ride the bus or metro or use taxis to get around. It is a larger city so walking all over is possible but tiring.
If you're leaving Siena to the latter part of your trip as you head back down toward Rome, I'd recommend you rent the car once you're back in Florence. Are you planning to stay outside of Siena and touring the countryside? Then the car is definitely needed. If instead you're just going to visit Siena, then you can get there and around without a car.
Car rentals do offer cars with automatic transmissions, you just have to make sure you request it from the very beginning as they aren't widely available even with the rental companies. If you have a US drivers license, contact the AAA to get the international driver's license - it is a document that accompanies your state driver's license. You need both to rent the car and to drive in Italy and Europe.
If you're leaving your visit of Rome to the end, you don't need a car there - whether you want to drop it off in Siena or Rome depends on whether you're returning "slowly" or "quickly" back to Rome. By slow, I mean you want to enjoy the drive down and the landscapes, stopping along the way. Then as part of your vacation it makes sense to enjoy having the comfort of the car to get to where you're going. You can drop it off at the airport and take the train shuttle into Rome to enjoy your last few days.
By quickly I mean a dash to just get back to Rome to then leave to the airport. If you're going straight to the airport to catch your flight, you could drive and drop off there but I am afraid the enjoyment of the ride lessens. Give yourself lots of ample time just in case you find traffic along the way.
If you're leaving the day after, I suggest you return the car in Siena or even Florence and catch the train down, sleep in Rome and then take the train shuttle to the airport.
You have a lot of options - depending on what your itinerary and flight schedule looks like, you have to see what modes of transport best fits what you feel like doing.
Let us know if you have any more questions, we're happy to help you sort out your doubts! Hope you're enjoying planning your trip
