You can certainly do the trip you have planned with the time you have. With say, 3 days in each of the 3 Cities and a 7 day drive, you have 5 days to do a little exploring on route.

I would like to take the trip to NYC by car so I could see as much as possible...
The one thing to keep in mind here is that covering a whole lot of miles doesn't always mean you will see more, other than the 'black stuff' and what you see view through the windshield that is. Sometimes, more is less, so to speak and I actually believe that you would see more following Buck's recommendation above. That said, if you want to drive across the country and stop along the way, then that's what you should do. You will have to pay quite a hefty one way drop fee on your car rental.

You have many routes and options open to you and you will need to look around the forums and road trip planning pages above, with a good map to hand and start to plot the things that appeal to you and try to create a route around the dots. One route from Vegas would be to head to the South rim of the Grand canyon and head through Monument valley and up through Moab [home to Arches and Canyonlands NP's] to I70 and into Colorado. Rocky mountain NP is close to Denver and there are many other options along the way. As you head east you could head for St Louis and it's famous Archway.

Have a look around and once you have some more dots on the map we can help to fill in the blanks.