For someone making their first cross-continent North American RoadTrip, I would suggest something like your Route 1: Follow the coast and the Appalachians into the south, head west through the Ozarks and Red River Valley, experience the Spanish Southwest, see the Red Rock Country of the Four Corners area, and finish up on the Pacific Ocean. I have nothing against the Midwest (My family has its roots there.) but it can be a bit plain vanilla for the traveler seeking adventure, and your Route 1 simply exposes you to more of the different geography and cultural diversity that America has to offer.
As has been mentioned, cities are not usually RV friendly, so I'd aim for those that either are or have good public transportation. In the former category would be the newer cities of the southwest that have broad multi-lane boulevards such as Las Vegas and Phoenix, even Santa Fe to some extent. I the later category would be older eastern cities such as Washington DC where you can park the RV well outside of town and take the Metro (subway, tube) into center city. Towns on or near your route that are well known for their nightlife would include Nashville, Memphis, Las Vegas and Los Angeles. But also check out smaller towns that are home to large colleges throughout the South and Central Plains, especially those that host SEC or Big 12 schools.
One other thing. You're going to be close enough to the actual Grand Canyon to simply include a visit to the National Park in your driving itinerary and physically stand on the rim. I think you would be sorely disappointed with a helicopter flight from Las Vegas. First of all, those flights do not go to the main part of the Canyon (Flights into the Canyon are forbidden.) but to the 'West Rim' which is on Indian land where the Canyon is not nearly as deep or spectacular as it is within the national park, and most of your flight time is spent getting from Las Vegas to the 'Canyon' with comparatively little of it being even in the vicinity of the Canyon.
And finally, at the first national park you come to, buy an annual pass. This will cost $80 but, similar to the National Heritage Pass, is good for admission to all national arks and monuments for one year. It does not, however, cover extras such as camping and concession fees.
AZBuck