The idea of a classic road trip is the jewel in the crown for fans of Americana - but it requires careful planning to execute well.
For a start, there's a fair bit of red tape. Car hire firms will hit you with extra charges for being under 25, and if you wish to drop it off in a different place to where you picked it up. Also, America is less of a country than a continent. It really is massive. NY to LA is a 5-6 hour flight. That's literally days on the road - just if you go straight from one to the other. You'll eat through money in motels, restaurants, petrol stations, and so on. You won't all be able to party as someone has to drive.
A good thing to do would be to concentrate on one area - I have a friend that went around the South, from New Orleans to the Carolinas via Mobile, Montgomery and Atlanta. Another drove down to the Carolinas from Boston, which is a bit longer. There's a lot of good stuff to see in concentrated areas, rather than shooting for the moon and doing NY - LA.