Here are a couple of places a pit in the country...
Sky Meadows State Park
http://www.dcr.virginia.gov/state_parks/sky.shtml
It is in Virginia's wine country. It includes a nice historic house and quaint, rolling hills. Also, part of the Appalachian Trail runs through the property, if you want to take a walk. It is just on the other side of "The Plains."
There is also the Virginia State Arboretum...
http://www.virginia.edu/blandy/
Similar to Sky Meadows, but with more rolling hills and stone country walls. The arboretum might be a bit more secluded than you want for a first date, though. Both places are in wine and horse country. If you are old enough, you might want to check out a vineyard while you are there.