When we were there, we just enjoyed walking about learning about the different contrada (neighborhoods). I found one which has the same colors as my Univ Illinois - blue and Orange, so I got pins and other tchochki from this contrada. It was also great that this contrada had won the Palia most recently. We also bought some of the special fruitcake from a bakery. We didn't climb the tower, as our climbing days are over. We actually did not make it into the Duomo, due to the line.
We enjoyed the bistecca Firenze at Osteria La Sosta di Violenta, a recommendation from a Sienese.