Hey,
Driving from New York to LA would take about 4 days of about 10 hours driving per day, without stops to see the sights.
What is worth seeing is quite a difficult question to answer. You are traveling pretty much the entire width of the country which means that you have everything to see in between!
But since you are traveling now, I recommend heading south where you will still encounter some warmer climates. You can head town to Tennessee and stop at the Smoky Mountains National Park. At this time of year you should see some of the nice fall foliage. http://www.nps.gov/grsm
If the outdoors aren't your thing but music is, stop at Nashville Tennesee or Memphis Tennessee. Both are well known music cities for Country and Blues. www.usaourist.com/english/places/tennessee/nashville.html and www.usatourist.com/english/places/tennessee/memphis.html
Further west, Santa Fe New Mexico is a good stop. Santa Fe is known for it's southwest architechture. It is also a haven for artists. A visit to the various galleries can find a great momento of your trip.
Not far from LA is Joshua Tree National Park. This park is popular among rock climbers but can certainly be enjoyed by all. http://www.nps.gov/jotr/index.htm
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