Is Renting a Car for Tuscany/Umbria Advisable?

If you rent a car, I would get it as you leave Florence. You are right, too many areas you can't drive. But, for the countryside of Tuscany and Umbria, it makes life easier. You can return the car in Orvieto, IF you rent with Hertz. That is the only place.
Remember you need an International Driving Permit, available at AAA for $20. Each driver would need one.

Yes there is gas, parking etc, but you have the freedom to come and go when you want without being dependent on a bus or chain schedule. Rental car, bus, or train is a personal choice. If I’m going to stop in any small towns, I’ll get a car every time. For large cities only, I’ll use public transportation. No need for a car in large cities. Once you have an itinerary set, use google to check for parking areas in the places you intend to visit. I go as far as to use street view to do a virtual drive by.

You really sort of need to have a decent idea of the places you might go, in order to make this decision. With a list (even if it's a bit too long to be realistic) you can go to the Deutsche Bahn website and check train frequencies and times.

I use only public transportation myself, but it often means that the best I can do is see two small towns in one day--even if they are so tiny that each really needs only an hour or two. But that is not necessarily a concern if you are mostly interested in somewhat larger places like Arezzo, Assisi and Perugia. I just always seem to have a bunch of little places on my target list, so I spend a good bit of time hanging out around bus stops.

Speaking of which: When is your trip? If you're traveling in midsummer, it can be pretty darn hot at those bus stops. I often think fondly of what it would feel like to be sitting in an air-conditioned car. (The buses themselves usually have a/c.)

Thanks for all the replies - very useful tips! I am traveling in early June through June 15. I will have to figure out which towns I want to visit and then decide on the car. It sounds like we can cover more territory with a car, though!

Larger towns (Siena, Arezzo, Pistoia, Lucca, Pisa, Perugia, Orvieto, etc.) are well served by public transportation and a car is generally not necessary. Also the traffic restrictions and/or congestion make the car somewhat unusable and you would still need to leave it parked at a garage/lot and visit on foot.

Smaller towns and villages are generally best visited by car because public transportation is often not as convenient and not that frequent. You can cover many more with a car than without one.