Two things come to mind right off the bat.
One, the (indoor) Mercato Centrale, specifically the bottom floor with the cheese & prosciutto shops, then the Nerbone stand for a great boiled beef with pesto or lampredotto sandwich. Should not be missed. Also, a rainy day at the Uffizi Gallery probably would be close to heaven for me.
Oh, if you're on via Tornabuoni near the Arno river, stop in (for free) at Chiesa Santa Trinita to see the Sassetti Chapel. Bring a couple Euro coins to illuminate the frescoes. The ones done by Ghirlandaio in the late 15th century are fabulous, better than any photograph.