I haven't actually done what you describe, though I've driven across the country many times --and I've ridden the rails for stretches of up to eighteen hours. So I only have parts of the experience to give you a definitive recommendation ...but I guess I can extrapolate.
Crossing America's heartland is pretty boring by car: miles and miles of dull scenery, awful sprawl and identical businesses. By train you'll be spared some of that, though you'll get neglected industrial wastelands instead.
Human interaction depends on your gregariousness and whether you're traveling with others or alone. If I were traveling alone, I'd regard the trip as a relaxing opportunity for introspection and reading --or it could give rise to an amorous adventure.
Will you rent a sleeping compartment? If not, sleeping in your seat is pretty rough. If you're traveling with kids, they'll vex you with their boredom.
Even if you're a paragon of equanimity, your patience will be tried by the hours spent motionless on the track or crawing at a snail's pace. This is because Amtrak trains are guests on rails owned by freight lines, who insist on prioritizing their goods. You'll actually sit for half an hour at a time waiting for a freight train to pass you!!
Consequently, on a long trip you'd be wise to expect arrival at your destination as much as twelve hours late, or even more. Consider yourself extremely fortunate if your train arrives no more than two hours past its scheduled time.
The trains themselves are reasonably comfortable --good seats, a fairly smooth quiet ride and HVAC that usually works-- but beware of the ones with the little plastic windows. Google up some pictures of the particular train you want to take (they usually have names).
Or try to get a cheap plane ticket. Southwest, Jet Blue... http://www.jetblueairways.com/cgi-bi...mp=minibooker&
Your time is better spent in New York than on a train.