It is always hard to design a trip for others; sometimes hard enough for just ourselves. Anyway.
So, twelve nights. Think about this:
Arrive in Venice, stay four nights.
Train to Cinque Terre, stay two nights.
Train to Siena, stay three nights.
Train to Rome, stay three nights.
If you more nights, add them to Rome.
Venice. It’s a great place to get over your jet lag, and fun to explore. Your older children would enjoy the night life in Campo Margerita, where the student population of Venice hang out of an evening eating pizza bought by the slice and drinking spritzes. Many will think I am crazy, but just ignore the big ticket attractions, the DK Eyewitness Top Ten. We have found in Venice that it is the “off Broadway” things that we most remember, the small details. Get hold of a copy of Secret Venice by Jonglez, maybe in your library, but otherwise it is worth buying. It lists many of the small things, details that you would otherwise miss.
Maybe do a boat tour with Gian-Luca, which we did half a dozen years ago. Link is here. https://www.likeavenetian.com/driving-a-boat/ The tour is really worth doing, particularly with a child.
Train to Cinque Terre takes about five hours, so if you make an early start from Venice, you still have a decent chunk of the day there. We visited the CT, spending two nights there; for us that is sufficient to get a handle on the landscape and the way agriculture works there, terraced olive groves and so on. We stayed in Riomaggiore (I think, in any case the closest town to La Spezia, the most southern town). On our first afternoon, we took a ferry ride up the coast to Manarola and returned on the local train. Next day we trained to Vernazza, walked to Corniglia and trained back to Riomaggiore. A child would find the walking hard going, and not that much fun. If looking for a beachside holiday, I would not pick the CT, although Manarola does have a beach.
Train to Siena from CT is about three hours, maybe train to Florence and bus to Siena. You could hire a car at Siena for maybe just one day, and take a trip into the country, or maybe just take a bus to a small town for lunch. Remember that parking in just about any Italian town is somewhere between a nightmare and a horror movie.
So a day looking around Siena, and a day in the countryside.
Train or bus from Siena to Rome, about three hours, for your last three nights.