Italian towns with good train connections

This post may have some info you can use
https://community.ricksteves.com/travel-forum/italy/trains-in-tuscany-for-smaller-towns

Renting cars is really crazy since Covid, I’ve not been able to find much under $100 a day, even in many US locations. In Italy I spent over €100/day for a compact few months ago. In your case, you have an even bigger challenge with 6 passengers. If you really need to rent a car for a few days, consider renting two economy cars (one per spouse) to accommodate 6 people. It will certainly be cheaper than one van, by far.
Check www.AutoEurope.eu and also www.AutoEurope.com (same company but check both sites because offers and prices differ).
In your case you can certainly save by minimizing your rental days and relying on public transportation for most of your needs when you travel to locations that are well served by trains/buses.
First of all I suggest you borrow one night from Tuscany or whatever and increase your nights in Venice to 3 minimum.
Rome: 4 nights is good. No car needed here.
Umbria: 5 nights. if you base yourself in Perugia there are several places you can visit by bus or train from there like Assisi, Spello, Gubbio, Spoleto, Orvieto (Orvieto is an easier day trip from Rome too, but you would need to spend an extra night in Rome otherwise you cut Rome short).
Tuscany: if you base yourself in Florence (Tuscany’s capital and transportation hub), you can visit several places without car. You callus also split your 11 night stay between Florence and Siena. Siena is only 75 min from Florence by bus, but is a good bus hub for small villages in its province (like Monteriggioni). The popular Val D’Orcia, south of Siena, would probably be best visited by car, but you can rent a car (or two) in Siena for a couple of days without spending a lot.
Cinque Terre: 3 nights. No need for a car here. Train is best (from Florence).
Venice: 3 nights. No need for a car here, unless it’s an amphibious vehicle (not allowed in the Grand Canal anyhow).
Dolomiti: 5 nights. Although there is the possibility of going by train and bus (train to Bolzano, then bus to Ortisei). However this is probably the only place where I would splurge in a car. Rent in Venice and head here by car(s). I would still base myself in Ortisei, if you want to stay in one place without hopping from hotel to hotel. If a car is not an option due to affordability then go to Ortisei by train plus bus and use it as a bus hub. There are several bus options available in Ortisei which can take you around the mountains.

You’ll probably want to reverify in just what order you want your route to take, but for Umbria, either Orvieto (west) or Spoleto (east) could be good. It depends how “deep” you want to get into Umbria, and if a daytrip to Assisi and/or Perugia is desired.

For Tuscany, on the west end of the region, look at Lucca (3 nights) then move to Florence for a 4-5 night stay. Lucca qualifies as “quaint,” and Florence maybe isn’t as quaint, but it’s an essential place. Look at rentals in the Oltrarno neighborhood, on the west side of the Arno River. Good daytrip offerings from both Lucca and Florence.

Emilia-Romagna, it’s Bologna. The train and bus stations are at the northwest edge of the old town.

I did five nights in the Dolomites without a car, and it was fine. We stayed in Ortisei (Hotel Grones— highly recommended) and never ran out of wonderful places to hike. There are three lifts just in Ortisei and you can easily bus to two other towns to access more lifts. Check the schedule to see what lifts will be open when you will be there. We were there in September.

Train connections are good in Umbria. We stayed in Assisi and loved it.

Looks like you will have a wonderful trip!