aside from the official holidays, most everything remains open
Ciao Michael,
Barberino Val del'Elsa is a very pretty town between Florence and Poggibonsi, I am sure you will like the Chianti area where you'll find yourselves.
Actually, the actual holidays in Italy are the 25th and 26th of December. Most shops will be closed the 25th for sure, many restaurants in Florence as well but many outside will offer lunch on the 25th with reservations (
read this post for some examples). If you have a villa, you'll definitely have a kitchen so you should just make sure to plan to have dinner on the 25th in to make it simplest.
Most museums actually are just closed the 25th and are open on the 26th, unless it were to fall on one of their normally closed days of the month.
Markets are usually kept going on regular days so you shouldn't have any problem in finding them on their usual days, make sure you get details from local tourism offices once you're here.
On New Year's Eve, many restaurants will offer a single seating per person feast.... just for an example, something like a 100 euro per person fee with dinner and party included up to midnight, including the champagne to celebrate the new year. You definitely want to make reservations for that ahead of time considering you're also a pretty large group for a special event. January 1 is also a holiday so everything will be closed, then so if you're still here for the 6th of January, that is a holiday as well, the Epiphany and there are several events connected to that day around Tuscany.
The last week of the year is generally a bit more quiet since it is right after Christmas but with New Year's coming up, generally shops and restaurants and museums remain open as usual. Florence, as any city, and the countryside as is Barberino, have their differences all year round so if you stay in Chianti, it will more quiet than staying in Florence... if you get lucky, it might snow like it has been these last day and you might get a white Christmas even in Chianti or Florence. If your family would like a day skiing in Tuscany, just head either to Monte Amiata sound of Siena or to the Abetone area north of Pistoia in the Apennines.
Hope I've been helpful, do let me know if I've raised other doubts

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