Time, Distance, and Options
Welcome aboard the RoadTrip America forum.
The route that MapQuest has suggested is one of several that you could take. There are others that are not too much longer, including I-76, I-81 and I-40. You could even modify that to branch off just past Knoxville and take I-75, I-59, I-20 and I-10. Have a look at a moderately detailed road map of the US and locate those three options and see what there is along each of them that might appeal to you. And those are just the 'backbones', you can always travel on roads other than the Interstates as long as you have the time.
Ah, but there's the rub. 4 days is barely time to make the drive from NJ to AZ assuming you do nothing but drive, drive, drive. No sightseeing. No detours to interesting cities or towns. With 6 days, you will have some time to do things, but only if this is a one-way trip. If you plan on getting back to New Jersey, even 6 days is simply not enough time to make the trip to Arizona and back.
AZBuck
Time, Distance, and Options - II
Sorry, but it's over 2400 miles from Trenton to Phoenix. That's 600 miles a day to try to make it in 4 days. It can be done, but as I said, it would mean doing nothing but driving all day every day. Even 500 miles per day (a 5 day trip) will get to be a grind relatively shortly. If you really want to enjoy the trip and see some things at less than 70 mph, then you're going to need to take at least 6 days. This would let you spend 2-3 hours outside the car each day, and if you spread that out through the day in a couple of stops, then you can end up seeing 10 or so attractions along the way and have a relaxed trip with some pleasant memories. Short changing yourself on sleep is NOT the way to go about this. It would just mean that you wouldn't have the energy to enjoy the things you've supposedly made time for, and would make you a seriously unsafe driver to boot!
AZBuck