Rome to Florence: up the coast then Florence; or central area to Florence

Are you flying home from Rome? If you’re flying round trip save Rome for last.
Take advantage of ITs public transportation and rent a car only places where a car is needed. Read up on ITs ZTL zone laws so you don’t incur hefty fines. A few seconds in a bus lane will result in a $300 fine.
On arrival day go to your furthest destination by train i.e. the Cinque Terre that would be a good place to recover from jet lag. The name of Rome’s airport train station is Fiumicino Aeroporto: https://www.trenitalia.com/en.html. I see a train departing Rome’s airport that requires a change in Rome with a direct train from Rome to Monterosso. You do not need a car in the Cinque Terre.
Visit Pisa the day you leave the Cinque Terre and travel to Florence. Store your luggage at Pisa’s central station and walk to the sights from there. From Pisa there are direct trains to Florence (Firenze SM Novella) station. You do not want to drive in Florence.
Rent a car the day you leave Florence which is the best way to visit Tuscany. You can drop off your car in Orvieto and take a direct train to Rome’s Termini station.

We need a bit more info
Where else will you be visiting and where are you flying home from?
Have you booked lodging? If so- are the bookings changeable/refundable?

You don't need or want a car in Florence, CT, Rome. A car is useful for the Tuscan countryside. I can't imagine having a car for 3 weeks in Italy- just not necessary and would become such a headache. YMMV
We have no problem renting and driving in Italy/Europe but we are also very glad when we drop off the car. On a 3 week trip we might rent/drive for a week of that trip and use the easy wonderful train system everywhere else.

Suggestion-
Fly into Rome- visit Rome now if you are not returning to fly out of Rome
Pick up car- drive to Orvieto- spend a night or 2, then drive thru Val D'Orcia and stay somewhere for a few nights to explore the area. Pienza and Montepulciano both make great bases and are fairly easy to drive in/out go and park a car.
Drive to Pisa- drop car here. Visit the Field of Miracles- all of the buildings are worth visiting- and stay the night if you like. You don't a need tour for this. The field is just lovely lit up at night.
Next take train to La Spezia and onward to your CT town.
Train from CT to Florence.

Where wil you go next?

If you are flying RT Rome then pick up your car on arrival and drive only as far as Orvieto (1.5-2 hours)- spend 2 nights. Great place to get over jet lag.
Driving after an overnight flight is risky - you know yourself best- will you be ok?, will you have slept on the plane? I wouldn't normally recommend it but in some cases it is what works best.
or
You could pick up rental car in Orvieto- but your only choice there is Hertz- AutoEurope doesn't deal with that location anymore-

Return to Rome at end of your trip and put all Rome nights at end.

It doesn’t really matter which way you go to Florence. That depends largely on what you want to see along the way. For example, if you are interested in seeing Capalbio, the Maremma area, Mt. Argentario, Massa Marittima, Pitigliano, Saturnia, etc. then the coastal route might make sense. If those localities are not in your radar, and you are interested in Orvieto, Assisi, Cortona, Val D’Orcia, etc. then it makes sense to go through the A1 freeway route, or even the Via Cassia (SS2).

What I would like to know is whether you intend to have a car while in Rome and while in Florence. I ask that question because this is your first time out of America and maybe you have not researched the “driving in Europe” matter enough, and maybe you haven’t even heard of the ZTL. Europe and America do not operate similarly when it comes to cars. Driving and having a car in big cities like Rome or Florence is a nightmare worse than having a car to visit Manhattan. Unless you are a resident with a City permit, cars are not even allowed in pretty much the entire downtown area of Florence or Rome, so I’m not sure you are even aware of that. If you plan to have the car in those cities, and even do day trips from there, you need to find accommodations away from the city center and preferably near the freeways, and rely on taxis or public transit to visit the city center.

https://en.m.wikipedia.org/wiki/Road_signs_in_Italy

https://en.m.wikipedia.org/wiki/Limited_traffic_zone

Thank you all for the great input - and great questions!

We are renting a car as we exit Rome - our next main stop will be Florence. The input about skipping the coast and heading into the central area is really helpful. We are now looking to stop in Orvieto on our way to Florence.
We realize many cities either don't have room/space/allow many vehicles, and we are definitely researching so we are prepared.
We'll be traveling to Cinque Terre after Florence.

Thank you for the perspective and info! :-)

The area between Rome and Florence is fantastic -- we stopped at a water fall park where we could sit in natural pools in a cascade of hot spring water -- free -- city park. Near Saturnia. https://www.discovertuscany.com/maremma/thermal-baths-of-saturnia.html#google_vignette. At the time we did this long ago there were no changing facilities and parking was tricky, but I had my suit on under my clothes and then just changed by the car leaving using my towel. It was one of life's more fabulous experiences although I smelled weird for days LOL. Sulphur water.

We stopped near Sovana and visited Etruscan tomb sites in the hills -- got a map in the town.
We also stopped at the Etruscan tomb cities of cervetari and taquinia just north of Rome and at several abbeys and towns along the way to Florence.

I have never driven in Italy. Just out of curiousity, can one rely on Google Maps to avoid the ZTLs? I usually trust Google not to steer me wrong.