Several posts criticizing the OPs choice of riding without offering up anything helpful about what he asked.
Steve, it pretty much depends on what type of riding YOU like. Since you live in the SW and have pretty easy access to the deserts and canyons on any weekend, I'd suggest spending time in the southest and maybe working your way north. Maybe spend a day at an Atlantic beach resting up the tired muscles after the ride out. Not a lot to see on the eastern coast, but just inland you can have a lot of fun weaving in and out of the mountains. I would suggest weaving back and forth across the Blue Ridge Parkway. Do some research on what sections of the BRP are best, and hit some of the awesome roads near it- Cherohala Skyway, Tail of the Dragon, hit the Wheels Through Time museum in Maggie Vallie, NC). Then take some twisty roads west across West Virginia.
What interests do you have? I'm a redneck at heart who likes old country music, so I would spend a day in Branson, MO, but most will laugh at that. Maybe hit the St. Louis Arch on your way through. Stay west if you want to see the plains. Locals find them extremely boring; as a New Englander who spent his life in the hills I found it quite interesting to see that much flat land. I wouldn't want to do it often, but I loved a ride across Kansas.
If that doesn't turn your crank, head south and wind through Arkansas. I hear there are some great roads to ride down there. Hit some old southern towns. Have some BBQ.
I hope to do a ride west this season, and will spend some time in the deserts and western mountains. The 50cc is a great way to get away from what you already know in a hurry, then you can ride back at your leisure. I'd probably do a 50cc out west then come back slow and easy, probably 400-800 miles a day with some stops at places I find interesting.