There's only one way to travel in the United States, and that is by car. Yes, you could theoretically use public transport, but that means you won't see much of what this country has to offer and what you'll be seeing is not the best end of it, either. Greyhounds and other transport tend to frequent only the most common routes, which means those economically important, not those visually or culturally interesting, and unless you only want to go from big city to big city they are only an occasional option. Most of the interesting bits lie far from the main routes, and if you want to see this country, you will have to get you own vehicle. This is also often true within cities, as American towns were constructed with cars in mind.
If you plan on traveling the USA the backpacker way, think twice. To the majority of Americans this mode of travel is an entirely alien concept and consequently there is virtually no infrastructure for independent travelers. Hostels exist only in big (or touristed) cities, and you'll find that they are either shabby or overpriced, very often both. In fact, getting a budget motel is often cheaper than a hostel and you are less likely to run into shady characters. Either way, the budget range is still around 40 $ a night, even for a dorm bunk bed (!) in popular cities, and you're not getting much for your money.
I reckon the best way to see the USA is by camper or even better, a car with a trailer, so you have independence from both public transport and expensive accommodation.
On the plus side, if you do have a car (rent it through a European site; often saves you about 50%!), America is very convenient. Almost everything you'll want to see has a well-paved road and a parking spot right in front of it, and driving here is literally a very straightforward affair. Be prepared to bring lots of caffeine though, as everyone drives along the totally straight roads on the exact same maximum speed which is the traffic equivalent of a lullaby. Also note that lack of proper driving lessons and a certain redneck attitude when on the road makes Americans some of the worst drivers on the planet. Road awareness seems to equal character weakness for many inhabitants, and you will need to make up for that with being extra careful.
But it's not all grim. Good food can be had, but it requires local knowledge and a well-fed wallet. Farmer's markets and food-conscious consumers are on the rise, but for now good food seemed like a white middle class hobby to me rather than a general ideal. In general I found that "American food" (Burgers, ribs, etc.) is the best taste for value, whereas all foreign food tended to be a lukewarm version of the original at twice the price, and mediocre even in expensive places. If you like good beer, however, do not despair. American microbreweries make some fine ale that is well worth drinking.
Food doesn't come cheap and you can expect to fork out around 20 $ a day (including tax and tips) on a backpacker's budget for food and drink. You can live off less, but then you'll likely be fat and slouchy at the end of your trip.
America's overarching social culture has both benefits and drawbacks for independent travelers. Where you come from matters little, and Americans are friendly, chatty and generally very helpful and happy to show you around. Except when they are unsure about your motivations or their surroundings, in which case they become excessively paranoid. You will find that in such cases, asking for the way will be met with suspicious glares or even increased pace and ignorance. Most of this paranoia is caused by America's media's incessant exaggeration of the various dangers in the country, and I cannot count the times I have been insistently warned of robbers, rapists and mountain lions, who always seem to be "in the vicinity" or "seen yesterday". Keep that in mind when trying to hitchhike.
As stated in the last post, Americans love to (or ought to) say 'yes' at all times, so don't assume any invitation to be serious and have a Plan B. In a very Japanese manner, social harmony is very important and sometimes the line between a good discussion and a unpleasant disagreement is very thin in America, where open disagreement is more often than not seen as a breach of social conduct rather than a healthy interchange of opinions.
Private sphere and private property are very important concepts in American society, and you better don't infringe on either of them. Once you are allowed within either of them though, Americans are gracious hosts in my experience and will go out of their way to make you comfortable. Americans love to party, and in big towns fiesta is never far away. To avoid cultural irritation especially if you are female, accustom yourself to the Bump'n'Grind, America's national dance, also known as the Backrubber Tango.
When I announced I was going to the States, many people asked me various version of the same notion: "Why? There is nothing to see." - "But, over there is like, you know, here." - "America? Why in the world would you want to go to that place?" For some reason there seems to be a sentiment that America is a place without much attractions, at least compared with more exotic places like Thailand or Brazil. Reality couldn't be further from the truth, and this leg of my trip has been one of the most impressive on my journey.
Another belief held dear by self-absorbed Europeans is that America is a country without culture or history. Well, if ancient cliff dwellings, Spanish and English colonial architecture, gold rush ghost towns and 300 years of human history of exploration, invention and entertainment aren't good enough for you, then Florence or Kyoto probably won't do it for you either. The American country has plenty of unique history to offer, and if you are willing to listen it will tell you your fair share of war and peace and human struggle, of alien gods and sunken cultures, and of heroes of renown. Many of this isn't necessarily marked (although mentioned) as a highlight in the big guide books, but well worth searching out.
In terms of contemporary culture, America has a lot to offer, too. I personally find that its appeal lies more in the various expressions of 'Americana' which scatter across town and countryside. If you stray from the (comparatively mediocre) standard fare of shows, entertainment centers and art galleries, America's contemporary culture is varied and diverse, sometimes bemusing and often openly hilarious. Where else would you find the largest ball of twine or the atomic testing museum? Where else could you tour an ethically aware porn studio or ride the world's largest rollercoaster?
Some of Americas museums, like the Denver Art Museum, are nothing short of excellent.
Yet America's most impressive sight is the country itself. From vast deserts and towering mountains to crystal-clear lakes and multi-coloured canyons, jaw-dropping scenery waits around every corner, sometimes within a mere hour's drive. Everything here is just bigger, wider and wilder than anywhere else on the planet, and America's nature has often left me speech- and breathless, as I looked around in disbelief. Even 'minor' sights, such as the Black Canyon of the Gunnison, will leave you gaping in awe. Photographs cannot truly capture the sheer grandeur of this country, and I can only recommend you to just simply go and experience it for yourself.
On the downside, if there is something to see, then someone will charge you for it, and that is true even for state-run nature sights. If you have to fork out 25$ per person to see a giant hole in the ground that requires zero maintenance, then you know Uncle Sam needs your dinero pretty bad.