When I was younger I lived in Fredericksburg, Virginia. The town has colonial history, many key early American leaders came from the area, and ties to the Civil War as several major battles (The Battle of Fredericksburg, The Battle of the Wilderness, The Battle of Chancellorsville, and the Battle of Spotsylvania Court House) were fought in the city, or near to it.
This is also the city that started nurse Clara Barton into forming the Red Cross. There's a Confederate Cemetery in the city. General Hugh Mercer was also from the area, so you can visit his old Apothecary shop. Another U.S. president, James Monroe, also comes from the area and you can visit a museum dedicated to him.
The city is close to George Washington's boyhood home Ferry Farm (later owned by General Hugh Mercer) so on American Independence Day my family and I would go there and have picnics on the lawn where our first President grew up.
Fredericksburg, is about an hour or so travel time from Washington D.C. And If you have the chance to go to D.C. I'd really encourage you to check out the Smithsonian museums.
A good road trip to me, is a combination of scenic splendor, local flavor, and finding sites of national significance. As such, you should look at the U.S. National Parks:
[link] which includes natural landscapes preserved for future generations, national monuments & memorials, historic sites, and battlefields. The link above, allows you to look for ideas by 'theme' and includes a 'civil war' option.