Hi Al and welcome to the forum!
Some initial thoughts?
Best area to base in Tuscany: I'd choose Florence, especially if train travel is going to be more comfortable for your member(s) with motion-sickness challenges. I have those too so anything involving in the backseat of a car or even some buses, depending on terrain, just isn't an option. Trains, however, were great and I've never once felt ill on any of them, either 'fast' or slower regionales.
Florence is a terrific transport hub for umpty day trips around the region by train or, sometimes, public bus. As many of the small towns and cities in the region don't allow one to drive right into them willy-nilly anyway, public transport works very well for the trips to the places most tourists are interested in (Lucca, Pisa, Siena, etc) And, of course, Florence itself is fascinating...or I think so, anyway. :O)
That said, there's a lot of love for Lucca so if looking for a smaller place to base that's on a train line, that might be an option?
You do not want car in Florence, and leaving anything of value in an unattended rental car anywhere at all is not advised.
Naples/Sorrento/Amalfi Coast: I would rearrange your itinerary to go directly to the region from Florence/Tuscany and put all of your Rome days at the end of the trip so as to be close to the airport of departure. You don't want to be in the A.C. region the night before an international flight from Rome.
Naples/Sorrento/Amalfi Coast: high-speed train to Naples is fine but THIS is the region where your companions with motion sickness are going have challenges. For all sorts of reasons - including lack of parking, ZTLs (if you haven't read up on ZTLs yet, you'll want to be doing that), and terrible traffic during high season - a car is going to be more headache than help in the A.C. in July, and the roads are VERY twisty. As most of the A.C. isn't serviced by rail, this leaves you SITA buses - not great for folks like me with severe motion sickness - or ferries. Sorrento is a good base for ferries to Capri, Positano, Amalfi and Naples, and you can also get there by (battered but very cheap) regional commuter train or less frequent Campania Express trains from Naples: the station for those trains is connected to Centrale in Naples, where your train from Florence would arrive. Those same trains also service Pompeii: only 30 minutes away by rail from Sorrento.
Luggage: if at all possible, I'd try to get down to no more than 3 suitcases: divide your youngest traveler's things amongst you +his/her carryon. You'll want to be able to move nimbly in and out of the trains, and travel in Italy almost always involves dealing with stairs so fewer bags - and lighter bags - is the goal. Pack fewer clothes and plan on doing laundry. Lightweight blends travel well, are comfortable in July heat (!) and humidity and can be rinsed out in the sink and hung to dry overnight. Plan on wearing every item you pack multiple times, and all pieces should be mix-and-match.
Rome, Florence and Venice are all safely, easily and enjoyably explored on foot (and some of Venice, of course, by boat). You may want to use some public transit in Rome for longest hops across the historic center but if you group sightseeing by area, you shouldn't need it much. We had one 4 night/3.5 day Rome trip where we didn't use it at all, and we didn't use it in Florence either.
