I really can't give you the tourist version of visiting a city. I live here and we don't research a place before visiting. We just go. I will give you my opinion and some advice.
Go to Rome. There is more for the kids to look at. That time of year is iffy with the weather. We have just come out of the rainy season, so there could stilll be some rain. Temperature is iffy too. Could be 60, could be 80. I suggest layers just in case. I also suggst you leave the shorts and tank tops home. Shorts are not worn by Italians and with so many churches to pop in and out of, you won't be admitted if you wear tank tops. A shame to get to the Basilica and discover you can't get in because you were wearing the wrong thing. Wear comfortable shoes. Rome is a lot of walking.
Food......Pizza here is very different from America. Since pizza was invented in Naples, what you get here is real pizza. The stuff in America is, well, altered. Most pizzas are rather small in size. Normal to order one per person if in a restaurant. Minus the crust, you can eat a whole one, because they are so thin. If getting a slice in a counter service place, one is enough. The crust is thicker and the pizzas are made much larger for those places to create the larger slice. Wood ovens are used, so a pizza takes about 3 minutes from placing an order to being done (if nobody in front of you). Sauce depends on where you are. Most places is just a tomato sauce. Really no serious seasonings. It tastes like stewed tomatoes. Some places might have some chunk pieces in the sauce, but most don't. The cheese is very spotty. Not alot on the pizza. It is made from the milk of a water buffalo. This cheese has a different texture.....sort of rubbery. It also has a twang to it. The cheese is made in my region, so it is not as fresh in Rome and pizza definitely tastes different in Rome. I prefer pizza in Naples for that reason. If you can find a pizza with smoked mozzarella, grab it! And as far as the local favorites, they like pizza with sliced hotdogs and french fries or corn, etc. Sit down restaurants put olive oil on the pizza before cooking it, so if you don't want runny pizza, ask for no oil. The kids won't get pepperoni pizza. Please don't ask for it. If you say "pepperoni", you will get a pizza with green peppers on it, because pepperoni means green peppers. The closest thing is a spiced salami, but when I say close, it is like saying California is close to South Carolina. Buy a rice ball. They are great. It is a breaded ball of rice deep fried. I have no idea what the stuff is in the center. It had peas and some other pasty looking stuff. I don't bother to ask. I just eat it. The fried mashed potato balls with cheese inside are fabulous too. NEVER ask for tap water. WHen you order water, ask for no gas and be warned that all wine is room temp.
Outside the Vatican is a hot dog stand. This is not an American hotdog. It is shaped like a hotdog, but does not taste like one and it will come on a bun that is more like a homemade bread shaped like a hotdog bun. Don't want to dissapoint the kids. Italians like mayo on hotdogs. They also like mayo and ketchup on the roasted corn on the cob the vendors sell
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Pastas are regional. For example, can't get tortellini or lasagna or ravioli where I live. Penne is the staple pasta. Someone else mentioned carbonara. Be forewarned that this is different based on who is cooking it. It is really good if cooked properly and is made creamy, but you could end up with a plate of oily noodles with some chopped egg and fatty ham pieces. This is not proper carbonara.
Sauce is different based on region. My region is more basic with tomatoes, basil, garlic and oil. Tuscany is more flavor with all the spices and meats. Don't expect breakfast to be what you are used to unless your hotel caters to tourists and has Americanized breakfast. Most Italians eat pastries and coffee for breakfast. I know the McDonald's in the Rome airport has egg McMuffins. My husband had a cow on his last trip when he discovered this. Said we needed to make a trip to Rome for breakfast
. Seafood dishes are different in Italy. Fish will come with heads. Shellfish will come with shells and shrimp will come unpeeled...heads and all. If you get gelato, just check the place to see if it is fresh. Some of the out of the way places might have older gelato. While the touristy places are more expensive, the rotation of stock is quicker and you will also have more choices. Some places give samples first. I know the place at the left of the Trevi Fountain does. If going to an American fast food place, you must pay for ketchup and mayo. If you have time, stop by the McDonald's near the Spanish Steps. It was voted most luxurious in the World. Oh, and there is a
Disney Store near there if you are interested. I found it by accident on my last trip.
Street vendors....It is instinct to reach out when someone wants to hand you something. DON'T! I find the Trevi Fountain the worst location for people selling crap. You really can't even enjoy the fountain because of these people. One walks away and another is on top of you with the same crap. They are extremely pushy. Just say no and the best thing to do is not even make eye contact. Ignore them.
For those with stands selling stuff such as statues, etc, don't fall for the "You are my friend. I sell you cheaper. I like you!" Haggling is allowed here. Do it! If you want sunglasses and they say 20Euro, tell them 10. If they say no, walk away. They usually run after you and say 10. You may run into some who won't budge. Just walk away. Never pay the first figure.
Theft.....Italy has a high unemployment rate the further South you go. All tourist areas have theft regardless. This is not America. Look at what the locals are doing. Women carry purses sort of in front, but have their hands on the strap. Men should never put a wallet in the back pocket. Always in the front. Nothing of value in jacket pockets. I can't stress this enough. Don't let a belt vendor come near your person. They will try to jerk your pants up and down to show you that you need a belt. They are actually doing this to distract the feel of their hand in your pockets to steal your wallet. The subway is a favorite for the pickpockets. If you have not followed this advice and are about to board the subway and discover your wallet was stolen just as you have stepped inside, no use trying to run down the person who did it. They do a subway car switcheroo. They more than likely handed the wallet to their friend and the friend has run in the car behind you
. Please don't give money to beggars. You will be sorry if you do. Giving means you are willing to give more. They won't leave you alone after that.
Toilets are usually cleaner outside Naples. However, always carry pocket sized Kleenex just in case there is no toilet paper. There is no toilet paper where I live.....no toilet seats either. Most decent places require you to pay to enter or they have an attendant. Give around 20 Euro cent. Places that don't have toilet paper in the stalls will have attendants who will give you an ample supply for your needs. Just carry your own. It is customary that if you use a bathroom in a small cafe, you must purchase something first.
If you go to a counter service place that has tables outside, you must order at that table if you want to sit down. You can't go to the counter and pay, then sit. Sitting means you pay more for the service. Just sit, order and they wil bring the stuff to you. Give a tip also.
Most restaurants add a service fee of a few Euro per person to the bill. You should still give a tip for good service. You must also ask for the bill. They won't rush you out of the restaurant. That would be rude. You let them know when you are done.
Carry your own medical needs. Cough meds, tylenol, allergy stuff, bandaids, etc. These items are not easily found here.
If Alitalia is still in service, find another airline to fly. Double check if you are flying Delta. Alitalia usually takes over for the Italian flights. For any airline, pack each suitcase with a variety of items for each person in your family. For example, suitcase#1, have a change of clothes for each person, meds, etc. Suitcase #2, the same. Luggage can and will get lost. It took my husband several days to get his back a month ago. Had he not been living here, he would have been without clothes etc as they lost ALL of his luggage. Italians are famous for striking. Keep your eye open the week before you leave to see if transportation is on strike. This means trains as well. Wouldn't want to plan a train trip from Rome to Sorrento and find you can't get back to Rome.
Don't walk in front of scooters and SMart Cars. They are the most unlikely to stop for you.
Check your hotel to see if they have AC...not all do and you might want it that time of year. You never know.
Most of all, enjoy your trip.
with Venice. We went to Burano and Murano (where Murano Glass is made) as well since we were there for 4 days, so we tried to see the other islands near Venice. I would go back there in a heart beat I loved it that much. 
