A week in Umbria in high summer + possible side trip

You mention one week, but it would help us give better advice if you could clarify how many full days you have, not counting your flying in or flying out days.

Sorry, I don't have a clear itinerary yet, but my spouse will be working during the week in Citta di Castello, so we will have a full seven days there, and I will be free to come and go as I like (mostly I will want to do day trips, but I am considering staying overnight in Orvieto and/or Perugia).

I only visited Orvieto in that region, but we absolutely loved the town. It's easy to get there from/to Rome by train in 70 minutes.

Italy's one of those places that's best traveled with a rental car rambling from town to town. You can also easily see some of the southern cities in Tuscany like Siena, San Gimignano and Volterra from there.

Ah! I didn't realize that Siena was within range. That's wonderful news! Is driving in Umbria/southern Tuscany difficult? I will be mostly on my own (see above) and don't speak Italian.

Thank you so much for the very helpful reply, MPH--Monti Sibillini National Park looks incredible! I will certainly get a car since, yes, pottering around is my chief intention. Thanks again.

There are too many options in the area but if you do go to Norcia ,and I think you should, you should also visit Spoleto and the Piano Grande. Spoleto has a spectacular location with a marvellous bridge over a gorge and wonderful cathedral.
I know you probably wouldn't have enough time but this is also a great area for walks/hikes and many tour operators offer walking holidays and day walks.

Here is a second vote for the Parco - Castellucio and the surrounding valley is just gorgeous. A shame you are not there late June when the wildflowers cover the floor of the valley.

My country home is not too far from the Val Tiberina there.
You need a car for sure. There is the Ferrovia Centrale Umbra, that can take you to Perugia but service is limited as it is mainly for commuters and many other places are not reachable other than by car.
Besides the towns you mentioned to the south, The gem in the area is the village of Anghiari. Arezzo is also very near and so is Caprese (Michelangelo's birthplace). Other than Michelangelo's birth house there isn't much more in Caprese, but there is a restaurant that specializes in porcini mushrooms and truffles that you shouldn't miss. Also nearby is La Verna, the Santuary founded by Saint Francis and where the saint received the stigmata. On the Marche side, Urbino isn't far and San Marino as well (although not an easy drive). Acqualagna is famous for truffles too.

David, Jkc, and Roberto, thank you so much for the wonderful suggestions! We would definitely like to walk while in Umbria, so we will investigate the local tour operators and see what's feasible. Thank you all again!

Does anyone think we should skip the side trip and simply stay in Umbria? This will be our first trip to Italy and we are not likely to return any time soon, so I had wanted to get a glimpse of Rome or Florence. If it weren't high season, I would be inclined to play things by ear and see how we feel, but given that we're talking July/August, I figure we should book all lodging in advance.

Rome is too far for a day trip from Citta' di Castello. It's a 3 hr drive each way. Longer by train.
Florence is doable, but only if driving a car, as public transport will take you significantly longer.
If you stay longer than just that week, you could visit Rome before or after your Umbria stay. Maybe you fly to/from Rome, I don't know, so you could visit Rome a couple of days before your flight back, if you have a couple of extra days.
If that week in Citta' di Castello is all you can stay in Italy, I would visit at least Florence on a day trip during that week. It's about 1.5 hr drive via the E78 to Arezzo and then the A1 to Florence.

Thank you for your advice--we are thinking of flying into/out of Rome and spending 2-3 days there. If that becomes unworkable, I will investigate arranging a quick, perhaps overnight, visit to Florence. Thanks again.

I have enjoyed Perugia, Assisi, Orvieto, and Gubbio all quite well. I have not visited Spello or Todi.

• Perugia is the largest of those, a hill-city, not a hill-town, and
  requires a fair amount of hill-walking, even after you take an
  escalator up from one of the parking lots.

• Gubbio was the quietest in my experience, very charming, with very
  few American travelers there this October, so I was glad I included
  it on my driving route. If you rent a car, then it
  makes sense to make more stops that are not served by train, while you have the opportunity.

• You'll also be within an easy drive of Urbino, with another Palazzo Ducale (more to see and built before the one in Gubbio, by the same duke). Consider lunch at http://www.anticaosteriadalastella.com/uk/index.php.

• Assisi can be quite crowded, but the people watching is good and the
  St. Francis history is significant.

• Orvieto's cathedral is one of my favorites, but since it's well
  served by train from the big cities, it would not be not hard to get
  to on another trip.

Thank you, Laura, for the immensely helpful post!