Overwhelmed mama needs some advice

5 places in 10 days (assuming that is 10 days + travel days from and to the US) is a lot to ask, and really means you'll only have 1 1/2 days (2 nights)in each place; since you'll lose about half a day moving from place to place- hotel to hotel.
Because of the difficulty getting go early morning flights out of Venice, consider flying into there instead. Book a multicity flight.
Consider staying 3 nights in Venice
Then 3 nights in Florence, with one day spent on a day trip to Pisa
Then the remainder in Rome. Spend one (very loooong) day doing a day trip to Pompeii.
Fly home from Rome.

Does your "about 10 days" include the arrival and departure days? If so, then you really only have eight days on the ground. Visiting all the places you listed can be done in eight or ten days, but it'll only be "drive by" visits. All your memories will be of train stations, train rides, and checking in and out of hotels.

I don't mean to rain on your parade, I really don't, but you need to seriously consider cutting way back. Since the kids really want to see Pompeii, that should be the priority. Think about flying into Venice. It is a wonderful place to get over jet lag and acclimate to a new world. Perhaps two or three nights in Venice, a couple in Rome, and a couple in Naples from where you can go to Pompeii. Then fly back from Naples. That will cut down on time lost back tracking.

The number of nights spent in each city should be determined by the amount of interest the family has in each place. One thing we learned a long time ago, is that it's not possible to see and do all you want on any trip.

I don't mean to be Debbie Downer, just offering food for thought.

Welcome!

First we need to know exactly how many NIGHTS you will have on the ground in Italy.
Is it 10 nights or 11 nights? Or is it really only 9?
Add as many nights as you can, your airfare is the same no matter how long you stay

A 3 night stay gives you just 2.5 days in a location. Each location change will eat up at least a half day.

It is easier to fly IN to Venice than out of -so if you can start in Venice.
End in Rome

Venice needs 3 nights
Florence needs 3 nights- add a night for every day trip
Rome needs a minimum of 4- more is better
Pompeii CAN be done as a day trip from Rome but it will be a brutally long day.

Suggestion
Fly to Venice
Venice 3 nights
Train to Florence
Florence 4 nights with day trip to Pisa
Train to Rome
4 nights Rome - I would not day trip to Pompeii- an option would be a visit to Ostia Antica just outside Rome
Fly home from Rome

You can see why you’ll want to extend your trip to as many nights as you can ;)
Save Pompeii and environs for your next trip

When searching for flights you want “ multi-city” or “ open jaw” usually under Advanced Search
You do not want 2 one way flights

Ok, ten nights on the ground, or ten days with travel? You don't have time for more than three cities. Perhaps review the RS guide book and suggested itineraries .
Switch your order, fly into Venice and out of Naples/ Rome. Venice can be hard to fly out of, especially with early morning flights.
The transportation /train from Venice to southern Italy will take most of the day. We looked into flying, but with getting to an airport early, security, possibly flight cancelations and delays, the train made the most sense.
Your type of question is coming up here 2-3 times a week now. Check the carefully -written posts under Italy and review Trip Reports posted here.
You are a bit late to be booking lodging for June, especially for four, but you could just get two hotel rooms. Still, you need to secure lodging asap.
Have a great trip and safe travels!

Thank you so very much for your input - it's a tremendous help!

I would fly into Venice.we did 2 nights there which was plenty . Arrival day rode the vaportto, walked thru city San Marco plaza rialto Bridge. Next day did tours. Slept and took the early train to Florence. 3 days Florence with one of these days a day trip .take early train to Rome or remainer of trip. Hit he ground running. Get the early morning tour of Vatican which will give you an early start and rest of day to tour. Go the golf tour at night..it was great

Three-six-nine, In and out of Rome. Golden Rule: no less than 3 night stays in any one place. Depending on your arrival time at FCO: Orvieto 3, Sorrento 3, Rome 3. I would not waste time on train travel with such a short trip. Nobody wants to spend a day moving. Not sure where you are flying in from but most big cities in the US will get you to Rome without too much fuss. Flying into Naples will require a change or two maybe even 3 from the US. There are fast trains from FCO to Naples. I have taken 3 kids to Italy for their 21st birthday, they will enjoy seeing one thing a day, that's it that's all. What they want to do is hangout, swim in the sea, play on their phones, walk around at night and see the buskers. Slow down, have some fun, see Venice on another trip and plan on at least 4 nights there. Pompeii is an all-day affair, have lunch in the cafeteria. Enjoy your trip I'm sure your going to have fun and make lots of memories. J

Is Pompei an absolute must? Because the other locations you're listing are not even close. If its a must, you can do this:
*Arrive into VCE, spend 3-nights
*Travel to Firenze, spend 3-nights
*Travel to Rome, for 4-nights
- dedicate a day for going to Pompei. It's a 2.5hr drive one-way so, it may be easier to get with a tour guide and let them handle the driving, route-finding and meals. This tour is one example that may take the stress-off this exception to your trip.
*Fly back home thru FCO
If Pompei can be skipped, you can include a night in Cinque Terre. It'll be a nice change-up to the cramped/hustle-bustle of the cities, let the kids stretch their legs and enjoy the coast.

1. Would you fly into Naples and out of Venice vs. one centralized airport?

Better to fly INTO VCE. As others have pointed-out, movement coming-into VCE is an easier affair.

1. How many nights would you recommend staying in each city overnight?

Given your rough itinerary, no more than 3-days, you don't have much time and you're getting into the American trap of trying to do too much in too short amount of time. Is this a vacation or a march?

1. Are there any other cities you'd recommend even as a day trip in this short timeframe?

Sienna or Lucca while staying in Florence. Maybe no day-trips and give yourselves a lag-day where very little is planned but, make it a leisure day. You have teenagers, tell them they're on their own for the day in Florence, make arrangements to meet for dinner or, afterwards.