Rogue,
Here's my take from personal experience.
Family trip, us, (grandparents), our 3 kids & their spouses, 4 grandchildren ages at the time 2/10 yr olds 2/14 yr olds. The time it takes to move this group is twice as long as moving as just a couple. I've also traveled with a group of 9 older adults. Complaints were different, but moving around from place to place presented some of the same problems.
Here's what I'd suggest.
1 N. Naples as you arrive so late in the evening.
The next day train to Salerno. Ferry to Maiori or Amalfi.
As noted above, Maiori has a good beach and ferrys to the other towns. It's easy to reach Pompeii by train from Salerno if that's on your list.
https://www.travelmar.it/downloads/orari_travelmar.pdf
3 N. AC.
Edit. This is where my daughter (who lives by Naples) and her friends stay.
https://www.simplebooking.it/ibe/hotelbooking/search?hid=3274&lang=EN&cur=USD&REFCODE=MOBILESITE#/q&guests=A,A&in=2018-11-23&out=2018-11-26&coupon=&REFCODE=MOBILESITE
Check booking.com using filters to narrow down your choices. For apartments and villas.
Return to Salerno for your trip to Rome.
4 N Rome.
Make sure you book for the Colosseum and Vatican Museums if their on your list.
Taxi to Rome Tiburtina station to pick up rental.
I use Europecar. You will have easy access to the A1 from here. If you've never driven in Italy, please read this and their PDF
https://www.italybeyondtheobvious.com/dont-mess-with-ztl-zones
Our g-kids enjoyed this park on the way to Tuscany. Pack a picnic, and let them stretch their legs. (About an hour from Rome)
https://www.italysbestrome.com/bomarzo-monster-park/
From Bomarzo head to an agriturismo, if this one is available, take it. It's stunning.
https://www.homeaway.com/vacation-rental/p6377683
4-5 nites Tuscany. Close by, (Acquapendente) very cute village, good market, wonderful cheeses, hams,
Wine and olive oil. They also have a blues festival in July and a art and food festival.
Things to do in the area.
https://saturniahotspring.com 1hr away. Everyone should enjoy it. Bring water shoes or flip flops. DON'T wear jewelry!! It turns black.
Another picnic spot, at least bring plenty of drinks.
Easy day trips. Pitiglino 30 min. Pienza, Montalcino and Montepulciano 1 hr. The whole of the Val d'Orcia less than 1hr. Radicofani, Vignoni, Castiglione d'Orcia. Could all be visited in a day.
Drive from here to Siena 1 1/2H or Castellina 1H 45m on your way to Florence. Drop car in Florence. Be aware of ZTLs.
2 N Florence. Should give you enough time to hit the highlights. Hotels are your friend here.
Afternoon train to Venice.
3 N Venice.
I really would avoid the CT. While they're some of the prettiest villages, they're also extremely crowded. For the time it would take to get there and the lack of things for the kids to do, I don't think it's a good fit with the rest of your itinerary.
If you have any questions, PM me.
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