I have in my past and present been a driving fool.
I can get from Virgina Beach to Denver in two days and from Denver to LA in 22 hours. I can do Denver, NY in 3 days. This doesn't mean anyone should, never mind all of this is done on boring interstate Highways. I much prefer secondary two lanes roads, the problem here is speed.
AS said above there are many parts of the states from NY to LA that are less then exciting. The whole of Kansas comes to mind, the bit of space between Grand Junction CO and just west of LA is best done at night, ya don't realize as much how much of nothing there is.
Then there is the truly truly beautiful bit of space from Denver to Grand Junction. As said above it can be a challenge for some cars, a friend of mine hated the drive in her Prius.
What time of year are you going? This will factor in as well. The drive though Kansas can be a nail bitter in the winter, WIND and SNOW! I70 is often closed there and between Denver/Glenwood Springs.
If you are doing this just to say ya did then my all means....have at it.
If you actually want to experience the states ya may be better off doing a bit more research and picking a region.
In June this year I did Denver north though YellowStone, Glaicer Nat Park, into Canada, West to Olympic Nat Park, South to Redwoods Nat Park all on two lanes roads, then put the peddle to the metal in Reno screaming along interstates, hit Canyonlands in Utah and cruised in to Denver 14 days from start. Mind you there was more diving then anything else. 14 days, 5 National Parks, 4000 miles and one driver. It was a hoot, I have pics to prove it.
I tell you this to give you an idea of how massive North America is. Place is HUGE!! 14 days and I just did a corner of it.
What ever you choose will be great fun.