I disagree about the cities thing. Some of the Best of America Are the cities.
Depends on your tastes.
Really, though, unless you are planning to spend several months, there is no way you can see it All.
What are you interested in? What is at the top of your list? Mountains? Coastal areas? Big open spaces? Culturally, what are you intersted in? We've got them all here! It might be easier to pick a general area/corner and explore it.
The greyhounds are cheap, and I wouldn't mind going shorter distances on them, but they are not going to be very comfortable. Especially if you are traveling alone.
I Do love taking the train. It is a great way to sit back and see the country-side. But they do often run late. (Amtrak (passnger trains) do not own the tracks and therefore do not have the right of way. You will get stuck waiting for freight trains to pass.) That said, IF you have a few days to spare, (It takes a Full 3 days continuous travel to get from one coast to the other by train.) *I* think trains are great. I think, however, You'd be disappointed if you're used to high speed, reliable European trains. =P
Amtrak used to have a pass where you could hop on and off for a month, or something. Not really sure how it worked. You certainly don't get a sleeper car with that. but If you're interested... amtrak.com ...see if they still offer that.
Honestly, without access to a car, you will be limited to what you can see/do. Unfortunately. America is TOO big. and most of it was built around cars. You Might be able to find large tour groups/companies to take you into the National Parks, etc. That may be the only way to see them if you do not rent a car. And even then, depending on how brave you are... I wouldn't go alone.