Welcome to RTA!
At around 1675 miles by the shortest route, which also happens to be a little on the scenic side, it's going to be at least a 3-day drive (doing about 560 miles per day). That's going direct with no sightseeing stops except what you can see out of the windshield.
There are loads of routes to consider, but here are the two "main" ones that use interstates:
1) I-76 to I-80, take that straight to Ohio where you pick up I-76 again, then to I-270 into DC. Pro: it's fast. It goes close to Lancaster. You can shoot over to NYC. Con: lots of tolls, through some of Illinois, all of Indiana and Ohio, and most of Pennsylvania.
2) I-70 all the way to PA, then drop down to Maryland on I-79 to I-68 east. This is a route that we just did back in '14, and it is really picturesque. Pros: Good road, straight shot, very pretty drives through OH, PA and MD. Dropping down to I-68 via I-79 avoids the toll road in PA, the PA Turnpike. Cons: Toll road through part of KS (not expensive in a car, tho). Does not go through Lancaster area.
There are also numerous routes to take you there via the two-lane roads. A good map or atlas should help you with this. If you're a member of AAA, you can get the maps free at your local office. If you're not, get a good road atlas from any bookstore or big box store, if you don't already have one.
Donna