I lived in the US in my twenties and travelled all over on the Greyhound with no ill effects.
As others have said, the main issue with Greyhound isn't the buses themselves, but more the bus stations. Many in smaller cities and towns are in dodgy neighbourhoods that aren't great to be in at night/in the early hours of the morning, especially if you're a lone female traveller, as I usually was. However if you plan your trip carefully, you can avoid being dropped off/waiting to change buses at times when you might feel unsafe.
People riding the Greyhound are from all walks of life, as with all public transport, and obviously sometimes you might find yourself sat next to someone you'd rather not be, but more often than not, you'll be next to someone just like you who's just trying to get somewhere without spending too much money. I loved riding the Greyhound - I met interesting people and saw parts of towns and cities I'd never have seen otherwise. If I were short on cash and had the time, I'd ride it again now.
Amtrak is great but expensive and limited in its reach to small sections of the East and West coast. The benefit of Greyhound is that the routes criss cross the continent and allow you to get to more remote regions and minor cities. If you want to do a proper road trip without driving yourself, Greyhound will definitely help you see the 'real' America.
However if you just want to travel between major cities quickly and easily, Greyhound probably isn't for you. Distances are long, there are frequent stops, and it isn't particularly comfortable being on a bus for several hours, as would be the case even doing a relatively short journey between two East coast cities such as Boston - NY. For comfort and ease I'd take internal flights or the Amtrak if possible.