Bob,
Which strategy to use will depend to some extent on which towns you want to visit in Tuscany and Umbria. If you're going to be only in larger towns well served by rail or Bus, then public transport might be a better option.
However, if you want to visit smaller "out of the way" places and have some freedom with your schedule, then a car rental would be a good idea.
As the others have mentioned, you'll need to be especially careful with "ZTL zones". Note that these are being increasingly implemented in many towns in Italy, so Florence is not the only place you'll need to be wary of these. I've seen them in the south also.
Note that you'll need an International Driver's Permit for driving in Italy, which must be used in conjunction with your regular license. Failure to produce may result in fines being charged on the spot! These permits are very inexpensive and can be obtained at any AAA branch. With the IDP, you'll get a chart showing road signs, etc. that you'll be encountering, so be sure you're familiar with those, and also parking signs and regulations!
I'd suggest a GPS unit along with a good Michelin map for "backup" (GPS units are not infallible). I normally get the full CDW these days as I dont' want any "problems", and trying to figure out the various coverage options with credit cards is a darn nuisance.
In the same circumstances, I'd pick up the car somewhere outside of Florence (perhaps the airport) and return it in Orvieto (from there it's only a short train ride into Rome). Note that the majority of cars will be manual transmission - if you want an automatic, you'll have to specically request one. Also, many rental cars use Diesel ("Gasolio"). Having a car will be good if you're planning to visit Civita di Bagnoregio, one of Rick's favourite hill towns. That trip can be done by Bus from Orvieto, but it's a lot slower.
Happy travels!