We will be in Rome in June and plan to drive outside (yes, I have driven all over Europe) Rome to some nearby hilltop towns. Any ideas?
I would go to Orvieto. It is 130 km from Rome and takes 1.5 hours to get there.
One more for Orvieto and it's close. if you'll have a car after Orvieto head west a short distance to Bolsena on Lake Bolsena. A beautiful little town on the small lake.
Even closer are the Castelli Romani, with Castel Gandolfo, Tivoli, and Frascati.
I vote for Tivoli too, and, San Gemini (Umbria), Spoleto, Montefalco, Todi.
Certainly Orvieto is worth a visit.
Orvieto and Civita di Bagnoregio. Both can be done in one day. Visit Civita first in the morning then spend the rest of day and night in Orvieto.
I agree with Zoe: Castel Gandolfo, Frascati, Tivoli are excellent choices and perhaps less-touristed than Orvieto where everyone with a Rick Steves book feels they need to go. At Castel Gandolfo you can take a tour of the Apostolic Palace. See the Vatican website for info.
If you decide to visit Orvieto, you won't be able to drive into town as it's within a ZTL (limited traffic) zone. As I recall, there's parking near the station and from there you can take the Funicular up to town. If you decide to visit Civita di Bagnoregio, there's a car park at the end of the pedestrian bridge that leads across to the town. I can't remember if it's pay parking or free?
There is much more to the region of Lazio than Civita di Bagnoregio or Orvieto (which is actually in Umbria). Explore beyond the couple of places Big Brother Rick Steves features in his guidebooks and documentaries.
Click on some of these very famous options all in the Lazio region, where Rome is located:
http://www.borghipiubelliditalia.it/en/borghi-details?view=region&id=7/
http://www.visitlazio.com/en/borghi-e-citta