For precise info, or if you are in doubt, it is important that you refer to the Government Guidance of your country of origin, and that you conform to your airline protocols.
At the moment of writing (February 2023) the use of face masks is not mandatory in Italy, and US travellers do not have to show their Vaccination Card to enter museums or other public amenities.
Tuscany Together, however, recommends the use of face masks in crowded areas like museums and trains, as well as the use of a face mask if you suffer from a cold or a sore throat.
On top of the governement guidance of your country of origin , feel fee to consult two Italian websites for detailed Covid-19 updates: