How much time does one need to explore Siena? We are staying there for 3 nights. During the first full day we are taking a tour up to Chianti and to San Gimignano (9-5:30) and looking to rent a car the during the next day during the day to explore other hill towns around Siena. So we will have our first day after traveling up from Rome, evening after the tour and late afternoon and evening of the second day and potentially some time in the morning after our last nights stay before driving up to Lucca.
Will the time free afford enough to see what we need to in Siena?
It depends, but as a general rule I'd say that, for the very basics, half day at least will be needed to visit the Museo Civico, the Palazzo Pubblico/Torre del Mangia (same location on Piazza Del Campo) and the Duomo/Baptistery. These have limited opening hours during the day, so you need to plan those around their opening times. You can enjoy everything else outdoor (like the Piazza Del Campo itself) in the evenings after you return back from your day trips.
Last year, I took a 2 hour walking tour with Siena Walking Tours [approx. $20.] It was very informative and interesting. I especially enjoyed hearing about the Palio races and the contradas. We walked around a lot of Siena during this tour, so I think it covers seeing what there is to see in the city, as it includes going inside the duomo.
That will be plenty, I loved Sienna but we were only there for 4 hrs or so and saw the major sites and what not. Bigger than most of the hill towns, so you can for sure spend all sorts of time there but I found it was enough.