Children are welcome throughout Italy, as Italians are extremely tolerant and patient with children. The majority of public transportation, museums and many other types of attractions offer discounted prices for children and students. It's always best to ask when purchasing tickets for child discounts. If you are residents of another EU community you'll find plenty of additional reductions available in state museums and other attractions. Unfortunately children and seniors coming from outside the US have to pay full price!
When dining out always ask for child portions or meals for kids "porzione da bambino" in Italian. As unlike many other countries you won't find children menu's advertised except at the local fast-food chain like McDonalds. You could also ask for a "mezza porzione" (half-portion) as a way of getting a reduced price for child or perhaps just ordering a simple plate of pasta that would probably be more than filling for the average child.
And wherever you travel in Italy you'll find popular child dishes like Pizza, Sandwiches and Ice-cream always available for them. Try to avoid soft drinks in restaurants and get your children use to drinking bottled water (sparkling is popular with kids). This might save you a small fortune as in a restaurant a litre bottle of mineral water will cost 1€, while a can of Coca-Cola could cost as much as four times that amount.
When travelling by train or other means of transportation try to visit a local supermarket prior to your travel to stock up on snacks for the journey, and save money at the same time. At many smaller grocery stores (Alimentari) as well as supermarkets you'll find a wide selection of breads, meats and cheeses to make your own sandwiches. With breakfast at your hotel, you can plan on just one meal a day at a local restaurant to save further money.
When travelling by train or bus always try to reserve your seats ahead of time, and if travelling by train always reserve places on a High-speed Freccia Rossa train when possible as they offer a superior service in comparison to the state-run trains in Italy. Always allow yourself plenty of time to get to and from train stations and other travel hubs. And whenever your budget affords take the easy option and utilize local taxi's instead of trying to find hotels in strange cities with your family dragging their luggage behind them.
The more you plan ahead, the more you'll enjoy your vacation. All of the major cities of Italy boast attractions that should appeal to your children, so don't spend every day in a museum. Try to vary your touring itinerary with plenty of surprises for your children. Also, get them involved in your trip planning and get them to study the places you'll visit. They'll appreciate their time in Italy a lot more when they visit places they are already familiar with from reading books, watching movies etc.