Self-planning an Italy trip and using trains is not difficult, especially if you did the "big 3"... Venice, Rome, Florence/Pisa. That's perfect for a 2 week itinerary. Just because you are in a city doesn't mean you have to spend all day in churches and museums. There's plenty to see & do just out walking around.
When you add Amalfi coast, Capri, or Cinque Terra I think it becomes a small bit more complicated because of transportation and time geting place to place.

Another place you might enjoy as a day trip from Rome or Florence is Orvieto. It's between the two cities on the train. It's a small city on top of a mesa, which you reach by taking a funicular up from the train station. Near the funicular station there's the remains pf an old fort that is now a park, which offers great views over the countryside, something you will not get much of in Rome, Florence or Venice. There's very few cars in the old town, so walking around is easy.

The one spectacular church, the Duomo, is beautiful, with the facade covered in mosaics that have a lot of gold in them. There is also an Etruscan museum across the piazza from the Duomo, although I don't know if the kids would be very interested. But the most interesting thing for them would be the underground part of town: first, near the funicular station at the top of the hill is a deep, deep well, dug hundreds of years ago to supply water in case of a siege. It's very big -- there's a double-helix spiral staircase that goes down to the bottom. There is an entrance fee. Then, some of the restaurants are located in caves, and there is a tour of Underground Orvieto, showing how many caves were dug in the rock and used for storage, workshops, etc over the centuries.

Thanks everyone!

alessandrazoe, wow, thank you so much for your very helpful advice :-)

Layanluvstotravel,

"Tours are planned for adults and for adult activities. You children would be bored with sitting for hours on a bus and not having any activities geared toward children. I would seriously discourage you from taking children on a tour."

Totally agree with Adrienne - considering the age of your children - they would be totally bored with traditional tours. Unless the tour is intergenerational with a focus on child friendly activities - forget it.

The OP now has another thread for his DIY trip.

I have taken a number of tours of various European countries with Tauck, www.tauck.com. I don't really remember children. However, they have a special series of tours called Tauck Bridges, which are specially designed for families with children. You could check them out.