That's a big trip, about six days each way, on the shortest routes. But there are a million ways to go.
Want to see Chicago? It won't add much more than a hundred miles to the trip, less than one additional day of driving time. But I can easily spend a week touring Chicago, it is one of my favorite places to visit, even after living there four years.
Or route through San Antonio, Houston, and New Orleans, for less than two extra days driving time, but each stop good for several days, much more if you want to explore Texas Hill Country or the Mississippi-Alabama-Florida Gulf Coast.
No specific suggestions, just trying to broaden your interests. I could spend 2-3 months getting from Washington to central Florida. I've been road tripping between NE Oklahoma and SE Michigan for 34 years and still keep finding new routes, new places to stop.
Washington to Florida is a huge world. If you have the time to make it a road trip, keep your perspectives wide. Mall of America? The Alamo? The Grand Canyon?
Indianapolis Motor Speedway? Saint Louis Arch? WJ Clinton Library in Little Rock? Opryland in Nashville? Home town of Popeye the Sailor? World's largest ketchup bottle (near St Louis) or ball of string (Kansas or Iowa?).
For this trip, any of this could be enroute with little variation in distance or travel time. It all depends on what interests you (which may not be what interests me) and how much time you have to stop along the way. If you don't have time to stop to see anything, just let a computer figure out the fastest route; that will be the same both ways.
