From Washington, DC, it is a little over an hour to Skyline Drive in Shenandoah National Park in Virginia. There you will see lots and lots of beautiful scenery, and you won't have to drive very far.
As others have said, there are many beautiful drives on the west coast. Two of my favorites are the Columbia River Gorge east of Portland, OR and California Highway 1 thru Big Sur (south of San Francisco).