I second the Route 66 idea. There are some great books on it. And videos.
I remember the Yellow Ribbon song well. That morphed into tying yellow ribbons around trees to welcome returning military service personnel. (Do people still do that?)
That is insane and, I wager, unsafe. I’ve done 18 hours non-stop once and, in retrospect, that was really stupid of me as I was popping Nodoz and chasing them with gas station coffee towards the wee hours of the morning. It’s hard enough to stay up 36 hours straight, much less do it while commandeering a motor vehicle and being responsible for one’s own safety and the safety of others on the road.
I did that once going in the other direction, back when there were only three trains a day and I had just missed the last one. It wasn’t bad besides the fact it was a slow trip because the bus stopped nearly everywhere. Another time I took Continental Trailways from San Bernardino to San Diego. Again, not too bad except on this occasion the bus smelled like cigarettes.
I did this twice, and would not recommend it for anyone between 25 and 65. You need a sense of humor and/or an excuse for a good story.
I also ‘did’ Old 66. If you drive, bring allergy medication, because no-name motels will trigger every allergy, and spawn new ones.
Seconded on Megabus. I have bought hundreds of $1 tickets between St Louis and Chicago - every single Friday STL to CHI, every single Monday CHI to STL. The thing is, I don’t use most of them, but buying and wasting them is cheaper than waiting until I need them. It only has one half-hour stop between St Louis and Chicago, so you’re not seeing all the tiny towns between, but it’s far more comfortable.
I only ride Greyhound when I am forced to. You will be sharing the bus with people who have been in prison as recently as that morning. Yes, I’m stealing that line from Larry Miller, but it’s true.
When I Greyhounded Pittsburgh to Sacramento, an old “cowboy” got on in Wyoming. Look up grizzled in the dictionary to see his picture.
His sister, who he he hadn’t seen in decades and was his only living relative, had sent him money for plane fare to come see her on her deathbed. He used the money for booze, saving barely enough for a bus ticket. He was filthy, smelly, drunk, but a really interesting guy. Hey had pints of Black Velvet secreted in his boots.
He shared his whiskey with me and when it was gone I gave him cash to buy more (I was 18). I never found out if his sister was still alive when he arrived.
That’s obviously the plot of a movie starring Robert Duval. Get busy and write it. 
I’m with those saying don’t. Not for that length. It’s either very uncomfortable, not because bus’s are necessarily, but the time is so long. Or you have to get a motel room then it isn’t cheaper than flying, which it might only barely be anyway.
Otherwise I’d push back a little on ‘coaches’ (one of those telltale words= foreigner 
) in the US being at all bad. I’ve found buses, Greyhound or other, fine for trips you can make in a day without transfer, when in a few cases it made sense. Like picking up or dropping off a car for one of my kids within the northeast. Otherwise I’d drive, like most Americans for distances where flying is too much of a hassle, which I find it to be even NY area (where I live)>Boston or DC, let alone similar distances to smaller cities in the region. Trains in this region are supposed to defy the US norm and be convenient, but I don’t see how they beat the bus, being significantly more expensive, unless it really bothers you to ride with a lower income clientele. But I don’t get intellectually edified talking to people on planes or trains, rather I always hope my fellow travelers will just leave me alone, and haven’t found that bus passengers are any more likely not to. And I don’t believe the actual risk of bus terminals or passengers crime wise is material nowadays.
There is no single ‘real US’ IMO, nor other countries either except maybe very small and homogeneous ones.
And that was the driver. Badda boom!
There are some high-volume routes where bus is a fine way to travel. Washington to NY is one I know well; the buses are clean and reliable, if a bit cramped. Some other routes have competitors as well and my guess is service might be better on those.
Amtrak will be circuitous, with at least 2 transfers e.g. Chicago and Washington. If you can spare travel time, though, those would be great opportunities to do a little sightseeing. More money than bus, but much much much more comfy, and it’s fun to strike up conversations with fellow travelers. There is a recent thread with hints for long train trips. Bedrooms are available but at a hefty cost. Aha, here’s the thread.
If you can get a good price on a one-way car rental that may be easiest and most flexible. Do allow for a night or two on the road and a little time to stop now and then. Using the US Highway system vs the Interstate will be a lot more scenic and they move well except when going through towns etc. Numbered highways in the United States - Wikipedia
In 2002, I bought a 10-day unlimited travel Greyhound bus pass and I travelled from Toronto to New York. Then every night I took the night bus from New York to D.C. (or vice versa) instead of staying in a hotel. You definitely meet some interesting characters at the bus station in the middle of the night. I thought about taking a trip from D.C. to Atlanta and back, but that would have taken practically a whole day of bus riding (lots of dinky stops) so I dropped the idea.
Saw a video today of a Greyhound driver falling asleep at the wheel and the passengers begging her to pull over. https://www.washingtonpost.com/news/dr-gridlock/wp/2018/03/23/watch-terrified-greyhound-passengers-plead-with-their-bus-driver-to-pull-over-and-sleep/?utm_term=.4d8f9d2a8adc