One day car rental in Florence

To enjoy your trip more, just take the bus from Florence to Siena. It's easy. Teh train is also available but the bus drops you off closer to the Siena historic center.

You don't want a car in Florence. See the many other posts on this site for why (use the Search box).
This is the standard advice that has been given on this site to the "car in Florence" question by those who have not had a car before in Florence.
Our resident Florence expert, Roberto, who used to live in Florence, gives the same advice: you don't want a car in Florence.

For a day trip into the countryside, there are pleasant tours ou can book in advance--they do the driving, endure the traffic and parking hassles, and absorb the financial liability of driving in a place where you don't want or need a car and in a location that has so many ZTL's and other challenges.

While having a car while visiting Florence is neither necessary nor desirable, however it makes perfect sense to rent the car for a day or two to visit the countryside. If by countryside you mean Siena only, then that is not the countryside. Siena is a small size city where traffic is restricted in the historical city center (just like in Florence), therefore the car is of limited use and one might as well visit it by bus as suggested by Kent.

A car is however useful to visit other locations, such as the Chianti and the many smaller villages of Tuscany, which have less frequent bus service and therefore more easily visited by car, which would allow you to visit many in one trip.

If you rent a car for over one day (e.g. 2 days), you might encounter the problem of parking the car overnight in Florence, which your hotel may or may not provide depending on where you are staying. The alternative in that case is a public garage or carpark, which however will cost you up to 20 euro a day. The public garage/carpark does not have to be outside the ZTL, as the garage operator can enter your car license plate in the White List of the City's website, so that no fine is issued for the ZTL incursion.

If you rent from morning to evening, then the parking issue does not present itself. However the issue might be the limited number of hours you can use the car. Car rental offices typically open at around 8am but close at 7pm Monday to Friday. They are open only in the mornings on Saturdays and closed on Sundays (that is the actual opening hours of the Europcar offices in Florence). The airport location is open longer (8:00 to 1130pm) but picking up at the airport costs more. Some locations (for example Europcar on via Forlanini or the Europcar airport location) allow to return after hours (however extra charges may apply).

Europcar has 3 locations in Florence: Borgo Ognissanti (city center), via Forlanini, Airport.
Hertz has 3 locations: Borgo Ognissanti (city center), via Sansovino, Airport
Avis-Budget has 2 locations: Borgo Ognissanti (city center), Airport.

The above 3 companies are the companies the consolidators below work with. I recommend you use the consolidators for better pricing.
www.kemwel.com
www.autoeurope.com
(they are part of the same corporation but check prices in both as they vary).

In terms of car pick up locations to go to the countryside south of Florence (Siena/Chianti areas), I would pick up from Borgo Ognissanti (in the city center) and return it to the airport if you return after 7pm (which you probably should to enjoy a full day). If you pick Hertz, you could pick up at via Sansovino, however you will need a taxi to get there, whereas Borgo Ognissanti is probably walking distance if your hotel is in the city center.

There is no fear of incurring into the ZTL while leaving the center from Borgo Ognissanti. "Candid" cameras (that fine you) are only at entrance points, therefore you cannot be photographed on exit.

No particular tips for driving. If you stay in the countryside, Tuscan roads are mostly 2 lanes, not too wide, and with plenty of curves since Tuscany is very hilly. There are also the freeways, but I presume you will want to leisurely drive around villages and countryside, not a fast drive on a 4-6 lane freeway.

Print this for reference:
http://en.wikipedia.org/wiki/Road_signs_in_Italy

I presume you know that you need an International Driving Permit to drive in Europe (you can obtain one at any AAA office in just minutes).

Thank you Roberto for this detailed info. I will check the links and print off all the information you have provided. You have been helpful on many aspects of planning this trip.