I spent three years living in Italy and enjoyed every moment. Wonderfully mad country and a great place to take your kids.
A couple of words of caution. Compared to travelling in the US, Italy can be a little bit crazy. Driving is chaotic at best and the roads leave a lot to be desired. It is also a fairly big place so my number one piece of advice is don’t try and fit in too much. The best part of Italy is sitting in a cafe and watching the world go by.
I’d start in the Riviera, somewhere like Santa Margherita (you can fly into Genoa) and spend a few days exploring the coast… Portofino, Cinque Terre etc…
Then spend a day driving down to Florence via places like Pisa (only point is to see the tower) and an old medieval walled city called Lucca.
2 days in Florence is fine. Book tickets for the Uffizi with an English speaking local guide and try and fit in the Boboli gardens.
Then head for Tuscany. Truly one of the most beautiful places in the world. Base yourself in a village like Greve in Chianti and have a couple of days exploring, visiting wineries and eating the amazing food.
From there head down to Rome via Siena (great place for lunch and an incredibly beautiful city) and the E80 coast road.
Rome is a big city but the history is incredible. It is like walking around a giant film set. The Colloseum is an absolute must closely followed by the Pantheon, Trevi fountain and Vatican.
For your week in the South I would base yourself somewhere on the Amalfi coast, like Positano. From there you are close to Sorrento, Pompei and Naples.
Enjoy your trip and happy to answer any question you have!