Itinerary

Consider that every move eats up the better part of 1/2 a day.
Rome2rio is a logistic planning website that I like. It doesn't have real time pricing or timing, but you can at least get a feel for what mode of transport works better for you. Many like a car for Tuscany and Siena may be a good spot for that. If you aren't versed in European cars, you need to take into consideration the comfort of the passengers as well as secure luggage storage which may mean something much bigger than a compact or budget car rental. Ensure you are versed in all the limited zone rules, standard shifting, CDW insurance.
Somewhere in your plan, I would think a day or two of down time for kids to just chill may be valuable. Letting them do something familiar (like sleeping in or doing a day of gaming) may be what they need to kick start their interest for the next tourist line up, church or museum. Given the heat, and that many places do not have AC, you might all appreciate some chill time somewhere. I have no idea if there are water parks in Italy, but I bet all would appreciate that for a day.
I also wonder if you have built in enough time at the start of your trip to deal with jet lag. What I am seeing is arrival in Naples, tour the city, then head to Sorrento all on day 1. Guaranteed meltdown of someone. It may not be your intention, but that is all I can see in the outline. Capri needs a day, Pompeii needs a day and Amalfi Coast needs a day - 3 sleeps doesn't cover that unless you plan to bus to Rome at the end of day 3.
How many days in Assisi?
Sorry to ask so many questioins and possibly appear critical, but with what is provided, I am not sure I get enough of a picture to answer.

First, I would spend more time in Rome, there are so much to see there. Also, two nights in Florence is not enough.

You can do all these places using the trains or combination of trains and day tours.

Driving in Italy is such a pain, so many people get huge tickets driving there. Many cities ban drivers without permission to even enter the inner city. I read on these boards of people getting tickets for going 2KPH over the speed limit.

I appreciate your input. I have a driver scheduled to pick us up from Naples airport upon arrival at 800am planning Sorrento check in for about 3--4 pm with hopefully quick bite for dinner and early bedtime. I have a tour of Capri scheduled for next morning at 9 am. I have no other tours booked for area as I wasn't planning on hectic schedule so it isn't a big deal if we cannot see it all. I agree to allow down time for them as I think they would get "burned out " quickly on seeing churches and history. Only one night in Assisi.

Have you read a guide book yet? First, you have way too many places on your list, and they are not in any practical order. Try to list how many nights are allotted for each place.
How much time are you allowing for Naples? You will be jet-lagged most likely, so your arrival day is not the best for a tour.

And, then, "three nights in Sienna,(transportation unknown ?and no set itinerary?) two nights in Vernazza, (transportation unknown?) two nights in Florence(David and Market) "
You don't want to stay in Siena, which is a bus ride away from Florence, go to Vernazza, then back to Florence. Please look at a map, it will really put things in perspective.

Bus to Rome for three nights. No, No, No! As far as transportation, most of it will be done by train. Very easy to figure out. You do not need a car unless you are driving through Tuscany. Any major city will be a hindrance.

Also, when you move from city to city, it will most likely take up half a day, packing, checking out, getting to train station, arriving in next city, checking in to new hotel. It is best to stay at least 2 nights in a city, 3 or more, depending where is better!

Do us and yourself a favor by listing out a day by day itinerary. Do this after you've investigated how you'll travel from one place to another, including the time to pack up, get to the train or bus station, and travel to the next location.

Also, having teenaged grandchildren, I know some teenagers are go-getters up at the crack of dawn and others think anything earlier than 11:00 is early. How are your boys?

Have you traveled in Europe, especially independently) before? I think you are being overly ambitious. Your plan strikes me as one exhausting mad dash. Also, keep in mind you'll be traveling when it is hot and every place you've listed will be incredibly crowded (especially AC and CT).

Since you're from Louisiana, have you been in the heart of New Orleans at Mardi Gras or have you visited Disney World in the height of the summer season? Your trip strikes me of being three solid weeks of that intensity.

To answer your questions.... Yes, I do have an itinerary and YES, have studied before planning this trip. My tour is based primarily on RS best of Italy Tour which is 17 days. Yes, I have looked at a map, familiar with travel area and planning. On travel days, we do not have anything planned other than travel. We only have one set tour in each city so I do not think this will be overwhelming.

I guess I am fortunate that my boys are on a schedule since birth, early to bed early to rise. Therefore, I do not think this sleeping will be an issue. We have planned this trip for over a year now so they are really looking forward to it! We are not going on a trip of a lifetime to sleep in hotels, we are going to experience the country. However, we will have breaks in our schedule that will allow rest periods.

I have traveled in Europe multiple times before and no never independently. I have researched the temps while we will be there and they are between 60-72 which is far from "hot " especially for us (being from South Louisiana) as we are use to heat and humidity.

I live in the heart of New Orleans and we attend Mardi Gras events annually and have been season pass holders at Disney for years with Italian ancestry- so I guess we do have lots of energy :)

I have researched the temps while we will be there and they are between 60-72

Not sure when in the summer and where you checked, but I’m pretty sure Rome and south will be warmer than that. Last time we were in Tuscany was in December a couple of years ago and the highs were upper 50’s-low 60s. I’ve been in Rome around the first of August when the high was 104.

I don't know if you've booked your lodging yet, but by changing your path a bit your trip will flow much better. From what I've been able to figure out I'm 2 days short of 3 full weeks.

Day 1. Travel.

Day 2. Arrive Naples. Naples city tour. Driver to Sorrento. 1H30m. Sleep Sorrento.

Day 3. Sorrento. Day trip to Capri.

Day 4. Sorrento. Possible Pompeii day trip.
Circumvesuvian, 40m.

Day 5. Sorrento- Naples-Rome.
Train...Circumvesuvian. 1H06m. Trenitalia.com,Train...1H17m. Bus option 3H45m.
Sleep Rome.

Day 6. Rome. Colosseum/Forum

Day 7. Rome. Vatican.

Day 8. Rome-La Spezia Train...3H35m.
Train...La Spezia-Vernazza Cinque Terre Express
15m.

Day 9. Vernazza.

Day 10. CT Express-La Spezia. Rent car drive
To Siena. 2H35m. Park @ the Stadio lot.

Day 11. Siena. Day trip Castellina & Montereggioni.

Day 12. Siena. Day trip to Pienza & Montalcino.

Day 13. Siena-Assisi. Drive. 1H50m. Sleep Assisi.

Day 14. Assisi-Florence. Drive. 1H35m.
Sleep Florence.

Day 15. Florence

Day 16. Florence-Venice. Train...2H05m.
Sleep Venice

Day 17. Venice.

Day 18. Venice.

Day 19. Travel.

Hope you have those 2 extra days.

Thank you for your assistance ! Greatly appreciated !