Family of 5 Travel Around Italy

Looking at your itinerary, I have to wonder why you don't put your Tuscany stay adjacent to Cinque Terre. You could pick up a rental car in Rome (e.g. airport) and drive to Montevarchi, then after your stay there drive to Polverara, then drive to Varenna, and finally drop off your car in Venice. Going from CT up to Lake Como and then back down to Tuscany doesn't seem to make a lot of sense -- but I imagine you have your reasons.

In any case, with 5 people and luggage I would say you'll want to rely on a rental car from the time you leave Rome until you arrive in Venice, because train tickets for 5 people would be a substantial cost. When you drive into Florence and Pisa be especially aware of the Zona Traffico Limitato (ZTL) or Area Pedonale no-car zones.
https://www.tripsavvy.com/tips-for-driving-in-italy-1547237

Cinque Terre (Polverara - need car for B&B off the beaten path): 3 nights

I am confused by this. The only Polverara I can find is a bit southeast of Padua. Why are you associating it with CT?

Have you plotted all your destinations out on a map?

I was confused by that one too joe. I wonder if it's fir the Venice lodging.

For your day trip to Florence, do NOT drive into the city. Instead park here and take the tram,
€1,50 into town. Easy access and €5 day.

https://www.parcheggiovillacostanza.it/en/

For parking in Pisa, there's a large parking area on Via Piava a few blocks from the Campo dei Miracoli. Much safer than the other two lots that are a bit closer.

Polverara is in the mountains above La Spezia, maybe 40 min from La Spezia. Although it’s a quaint mountain village in the middle of a forested area you will sweat bullets trying to get in and out. The road is narrow and full of switchbacks. You shouldn’t consider that the typical Cinque Terre experience, because you won’t be anywhere near the sea. It’s really a secluded mountain destination. Plan in almost one hour to get down to Riomaggiore, I hope that whoever drives is comfortable with negotiating the car in very narrow spaces. It might be faster for you to drive to the La Spezia Central station, park in the multistory car park there, then proceed by train to the Cinque Terre.

Your itinerary should be:
Rome
Tuscany
Polverara (Spezia)
Lake Como
Venice

The way you have it it’s major backtracking back and forth. That’s crazy.

Based on your destinations, you should rent a car at the end of your Rome stay and keep it all the way until you arrive at Venice (Piazzale Roma is where the rental companies are, just inside the City parking structure).
You will need a large car or minivan to accommodate 5 people plus luggage.
Check options on
www.autoeurope.com

For sure you need to revise the sequence of your itinerary as I listed above, otherwise you will double the mileage you are covering.

I don’t know what attracted you to Montevarchi. Is it because you plan to spend 3 days at the Prada outlet there to buy shoes? I can think of a whole list of Tuscan towns that are prettier than Montevarchi, even in the same general area. You’ll have a car so you won’t be limited by public transportation options. If you want to stay near Montevarchi the village of nearby LORO CIUFFENNA is a gem.

Thank you for your advice on transportation. Thank you also for your thoughts on itinerary, but we have various reasons for the order of itinerary so do not need/want to adjust this. We understand that we are doubling back in the middle of the trip.

I edited our Cinque Terre location - the town name on our booking was showing up in an odd place on the map - it is actually 4 miles from La Spezia, but 30 minutes and a tough drive. I will keep looking for closer options to CT, but given budget constraints and a family of 5, this one is the best so far.

The Tuscany agriturismo is near Montevarchi, but closer to Pestello.

Renting a car at the end of Rome and before Venice sounds like a great way to go...but wonder if there's a good train option for one leg of the trip between Rome-CT or Tuscany-Venice so we can experience that?

If you change your Cinque Terre accommodations from Polverara to one of the 5 villlages, you could take a train (all 5 villages are served by rail).
You will need a car for the Agriturismo in Tuscany, so you would rent a car in La Spezia (near Cinque Terre) and drive from there.
If from Cinque Terre you go to Vsrenna, you can take the train (Varenna is served by rail).
From Varenna to the Agriturismo near Montevarchi, you could take a train to Florence and rent the car in Florence. You could even take another train further to Arezzo, and rent a car in Arezzo, to avoid driving in Florence.
If you need a car with automatic transmission, then I recommend Florence, because Arezzo has small rental offices where the car fleets offer less choice.

Here is a random sample of the route La Spezia to Polverara

Lot more interesting than a boring freeway, but looks time consuming, although a pretty drive

https://www.google.com/maps/@44.156299,9.790651,3a,75y,90t/data=!3m6!1e1!3m4!1sty8CIyXmJ9qrjVRo5vRTTw!2e0!7i13312!8i6656

Some people recommend Levanto, north of Monterosso al Mare when accommodations are tight in the Cinque Terre, or as a bit less expensive. It is a few minute train ride between them.

There are many trains between Roma Termini and La Spezia Centrale. Use https://www.trenitalia.com/tcom-en to look at schedules. there's a direct train (no changes) at 9:57 arrives 14:03.
Driving would take longer. Price if purchased early is showing €16,90.
From La Spezia regional trains run to each village.

Edited for link.