Monthlong Italian Trip - transportation questions

Take trains except rent a car in Florence to stay in towns like Siena for a few days. Then return car to Florence, train to Venice. For the vehicle lover, Maserati plant is in Modena near Bologna.
We took a day trip from Venice by train to Bassano de Grappa with it’s WWII history of the Italian Alpini mountain troops. There are memorials and a museum and a beautiful Palladian Bridge. Very interesting.

All the places you mention can easily be done by train and a car would be more of a hassle than help. The only places a car is helpful is in very small towns (Siena is a small city and linked by both bus and train to Florence) and some quite rural areas. I've been to Italy 14 times and only rented a car 4 or 5 of those trips (and even at that for only part of the trip).

I suggested the trains as you seem to want to be city focused. We have been to Italy ten times for three to four weeks each trip. We have rented cars all except the first trip when we mixed trains and car rentals. Look at US based AutoEurope for car rentals.
We have picked cars when leaving Rome, Milan, Florence and other cities.

For the places you mention, Rome, Firenze, Venazia, Cinque Terre, a car is very much a nindrance. Trains are the way to go, blast through the landscape at 250 km/hr while you browse the free on board wi-fi or enjoy a drink in the bar car.

Get the app Rome2rio, and plug in start and finish points. You can compare various modes of transport, car, rail, bus and so on.

It's dangerous to depend on Rome2Rio.com for detailed transportation information. It's useful (usually) for indicating which trips can be made by train and which can be made by bus. Transfer points, when shown, are generally correct. Unfortunately, you absolutely cannot trust the fares, travel times, or frequencies shown on Rome2Rio. For serious planning you need to go to the website of the transportation company. Rome2Rio will give you the company name and often a link to the website. That's where you can usually get correct information. For obvious reasons, the information currently displayed may not accurately reflect what will be happening when things have settled down and you can actually make your trip.

Rail schedules can be found on these two websites:

Trenitalia (for both express trains and local trains)
Italo (just running express trains to select cities)

Th railway system in Italy connect all the main cities and several towns and is in general quite reliable (at least about bullet and long distance trains). So if your idea is to stay only in main cities and touristic areas the car isn't necessary: you save a lot of time and stress moving by trains.
You can use a car to do daily tour in the countryside and are beautiful everywhere in Italy.
About cars one of the best area is Emilia-Romagna, where are located some of the most prestigious car factories (Lamborghini, Ferrari, Maserati, Ducati, Pagani...) . Check this website as reference: https://motorvalley.it/ . Other two important cities for cars are Milan (Alfa-Romeo) and Turin (FIAT - FCA group): in Turin is located the biggest Italian car museum.
For WWI the best area is the north east of Italy, means Veneto, Friuli and Trentino. There where fought the main battles so there are still a lot of memorial and fortress. The point about visit fortress is that are located in secluded areas, usually can be reached only walking for a long time on mountain hiking paths. Someone are only for skilled hikers. In Rome you can visit the Altare della Patria, the biggest memorial about WWI and now about Italian troops.
WWII was fought everywhere in Italy, so almost everywhere you can find memorial. There where a lot of battles in the mountains between Tuscany, Emilia and Romagna: the so calle Gothic Line. There are a lot of little museums here and there (Montese, Castel del Rio for example) and of course War Cemeteries (the most between Rimini and Bologna). These museums are more difficult to be reached, because are villages: in this case the car is mandatory.

So if your idea is to stay only in main cities and touristic areas the
car isn't necessary: you save a lot of time and stress moving by
trains.

Ditto to what Ricky stated above: a car is far more headache than help in the cities - you won't be using it to get around - and even in some other locations such as the Cinque Terre, or the Amalfi Coast during high/shoulder seasons. You should read up on the pedestrianized places you are not allowed to take a car, ZTL's, parking issues (there appears to be lots of those in Siena), etc. Hotels located in ZTL zones need special permission to access just to unload baggage before moving the car to a non-ZTL location. Anyway, for a first trip to Italy, there are SO many places that are easily accessed by train that I'd focus on using the rail system.

A rental car can be useful for exploring, say, the Tuscan and Umbrian countrysides although there will be hill towns you won't be allowed to drive into: you park outside of town and walk or sometimes take public transit in. Still, with advance reading, it can be a good way to get the feet wet regarding Italian traffic ins-and-outs.

Florence is an excellent city to base in for easy day trips by public transit. Within inexpensive reach are Siena, Fiesole, Lucca, Pisa and some others. Some travelers have day-tripped the Cinque Terre from there but I do not recommend it; the golden hours in the CT are the early mornings and evenings when the hordes of day-trippers aren't around so spend at least a couple of nights in one of the 5 villages (my vote goes to Monterosso.)

Tivoli and Orvieto are popular day trips from Rome but there's so much to see just within the Eternal City that you could spend many weeks there without running out of things to do! What I wouldn't do is day-trip Naples and vicinity from Rome: there is enough ground to cover in that region to warrant staying in Naples or along the Sorrentine or Amalfi Coasts for 5-7 days.

Our favorite experience so far was taking public transport out to a further reach of the Appia Antica (Casal Rotundo) and walking it backwards into Rome, along the crumbling remains of once-grand tombs and emperor Maxentius' private circus, a wealthy villa's private baths, catacombs and other interesting relics. Also along the way (for your WWII buff) there is Mausoleo Delle Fosse Ardeantine: the tomb of 335 Italians massacred by the Nazis in 1944.
http://www.mausoleofosseardeatine.it/home-eng/

Also related to WWII, look for brass "Stumbling Stones" set into the pavements around the Jewish Ghetto and some other areas of Rome.
https://www.thewingedsandals.com/romes-stumbling-stones/

All good advice from the others.
Not to beat this to death, but for the trip you propose having a car would (for the most part) be more of a hassle than it's worth for all of the reasons previously stated. To put it bluntly you don't want to go anywhere near any of the major cities with a car.
Since you'll mainly be connecting cities the train system is the way to go - will minimize the aggravation as you move around the country.
The only exception I would suggest would be to consider renting a car for a few days if you decide to base yourselves in Siena, from which you could explore some of the off-the-beaten-track countryside in Tuscany and Umbria. Am thinking particularly of the area around Montepulciano which was a highlight of our own trip last year. Motoring around the Tuscan countryside and exploring the Val D'Orcia at leisure is truly one of life's great pleasures. You could do it on an organized tour as well, but our preference has always been independent travel at our own pace. Might also be a pleasant break for you after the bustle of the cities.
Siena would also be a handy base for exploring Chianti, especially the scenic Chiantigiana road (the SR 222 on the map) that extends north of Siena. There are too many pretty little villages to count, either for a lunch stop or just to walk around and soak it all in.
I suggest Siena as a base because it's easy to get to via train or bus and once there collecting a rental car is a snap - there are a couple of places within a block or two of the train station.
I second the suggestion to use AutoEurope for your rental. Spring for an automatic (the default rental in Italy is a manual) and get the zero deductible insurance cover for the peace of mind.
And I agree that walking or riding a bike along the old Appian Way would make for a terrific day out.

Thank you everyone. I’m writing down every suggestion. They are wonderful.