How exciting! Lots of great suggestions here. I taught in Italy last summer and don't have too much to add. I always tend to dress up a bit, skirts, loose top & a scarf. Linen & natural fabrics. I wore black with colorful scarfs & fit right in.
Flat sandals are very popular, all colors and especially metallics. Lots of Birks--even the italians wore them. Sneakers were also popular, usually a classic leather style. A cross body or satchel bag big enough to hold water, book, camera & scarf. I would often wear a sleeveless top + skirt & carry a linen or lightweight blouse to cover up as necessary--August sun is brutal
For museums, I would reserve the Uffizi in the morning, and then later the Accademia as it is smaller and takes less time to visit. There are so many sights, the Bargello and Galileo museums are fantastic, as are the Boboli gardens it just depends on your interests!