You do NOT want to drive in Florence, trust me! Between the narrow one-way streets and the motor scooters weaving in and out, it is extremely difficult. If Florence is not your destination for that night, why would you go there just to rent a car?
It is very easy to rent the car in Livorno; however there may be a drop-off charge when you return it to a different city after your trip.
You will love Siena!