Venice Florence Rome is a classic first trip for many reasons. Fly into Venice, if you can, but if Milan is much less expensive it is easy to fly into Milan and immediately take the train to Venice. Venice is a terrific place to get over jetlag, partly because you can wander everywhere outdoors and never have to worry about traffic. If I were you, I would consult with my kids and perhaps add a day or even two to Venice --- the first time we visited Italy with our sons we stayed in Venice for a week (no day trips) and it was by far the favorite part of the trip.
On our 4th trip to Italy, my husband and I spent a week each in Venice, Florence, and Rome (no day trips) in December and January. We thought the weather was great --- it was cool and foggy in Venice, a bit of snow in Florence, and I seem to remember a tiny amount of rain in Rome. Anything is better than the heat and crowds of summer in all three places. If you encounter colder or rainier weather, these three cities have so many indoor things to do that it's not a concern --- you're not trying to hike or go to the beach or see natural scenery. The only winter issue I see is that for avoiding COVID, eating outdoors would be better. Last December, when we were in Umbria and Rome, people had to show proof of vaccination to eat indoors in restaurants and we really liked that --- maybe Italy will return to that for your winter trip.
There will be some restaurants closed for the holidays, and some museums closed for Christmas and New Years, but that is easy to work around. Book your lunches and dinners ahead of time for those days because people will be eating out. The one unexpected problem we ran into was that we strategized that January 4th would be a good day to visit the Vatican museum --- it would be open and well past New Year's Day so maybe it would be less crowded. What we mainly wanted to see was the museum rooms and art gallery --- I'd been to the Sistine Chapel before and my husband was not especially interested in seeing it. We entered as soon as the doors opened that day (more strategizing) and it was only then that we found out that literally ALL the rooms we wanted to see were closed because most of the staff were still on vacation. What we did not know (besides that such a thing was even possible at a major site) is that January 6th is the Epiphany holiday in Italy and so people tend to take off from work until after the 6th. So, do watch out for that. As for being crowded, it was was absolutely jammed (and I mean full body contact) with visitors all rushing to see the Sistine Chapel. Maybe splurge on some special after-hours tour to save your family from that.
I don't know if you are thinking about daytrips from any of these cities --- your time is short and there is plenty to see and do without any extra trips. The only two places you might add on with very little hassle or time spent in transportation (and much reward!) are Siena from Florence, and Orvieto from Rome.
Here's another "if I were you": traveling with family, we much prefer to stay in apartments so that people can have their own bedrooms and ideally there is a living room to relax in and a kitchen. Staying all in one room or even in a suite gets old real fast. Some will advise you to stay in hotels for your first time in Italy because the front desk will help you with things, but we have found that with a carefully chosen Airbnb apartment with an English-speaking superhost with lots of excellent reviews, the host is actually far more friendly and helpful than hotel staff.
Your family will love Italy! It sounds as though you plan to take it easy and not drag your family all through Italy at a fast pace in the middle of summer like my daughter-in-law's family did when she was a teenager. She does not have fond memories!
