Urbino-- can you rent a car there to drive into the countryside?

Try the Europcar website. Your parking option may be at the bottom of the town. You could probably rent in Pescara.

No major rental company in Urbino that I could find. Maybe local independents.
However Europcar is in Pesaro and I think Avis too. I would rent from there and have a car for you entire stay. Urbino itself can be visited in less than a day, but there are a lot of nice places to visit in the Marche region and having a car will make it much easier.
Use a consolidator like www.autoeurope.com or www.kemwel.com. It's often cheaper to go through a consolidator rather than directly with the rental company.

Thanks. I was hoping to use Urbino as a base for a few days. I know it's small, but I like to sometimes settle into a place for a few days to enjoy the local flavor, etc even if I'm just using it as a base. And, for what I was planning on doing I could avoid the major motorways.

You can still have Urbino as a base.
Rent a car in Pesaro, drive to Urbino.
Find a hotel in Urbino that has car access and parking (most likely it will have to be outside the historical center's limited traffic zone).
Stay in your Urbino accommodations as long as you like. Use that base to visit the region.
The only major motorway in the region is the A14 along the coast and the E78 from Fano to Urbino. If you are used to driving on a freeway in your country (I presume you are from a country that has freeways), driving there will be exactly the same. Just stay in the right lane and use the left lane for passing only. No cruising in the left lane of freeways is allowed (only passing).

Thanks again. I'm from the US. While I use the left lane on freeways only for passing, many of my fellow countrymen do not :-)