Sounds like a great trip. I rode from Maryland to Arizona last September and had the time of my life. For a June departure, I would be a little worried about the heat in the southern third of the country -- having lived in central and southern Arizona, getting enough water to stay hydrated on the road isn't easy, especially given how far apart the watering holes are. Take a bunch of water with you and stop frequently to drink up.
On your way from FL toward DC, I would suggest riding the length of the Blue Ridge Parkway and Skyline Drive -- cooler, less traffic, and entirely lovely. Heading west from DC, I went through West Virginia (the eastern third of the state has some great riding), Ohio (stop at the Motorcycle Hall of Fame in Pickerington), Indiana, Illinois, Iowa, and then across the fascinating and desolate Lincoln Highway (US 30) in Nebraska. Colorado has some awesome riding -- stay in the mountains and you can't go wrong. Southern Utah and northern Arizona are lovely, too -- US 89A through Flagstaff, Sedona, and Prescott is great.