I broke up with my long term GF and I now want to do something to forget the pain.
I decided to travel to the USA. I’ve just applied online for the Visa Waiver Program and got accepted.
Where should I go? I will be alone. No Vegas please, I hate casinos 
That’s way too broad a question. What are you interested in? Scenic natural beauty? History? Fun tourist attractions (like amusement parks)? Big city culture (shows, restaurants, night life, museums, etc.)? Quirky curiosities?
It might help us if you’d suggest things you DO like to see and do. Also, how much time do you have for your trip, and what’s your budget?
Also, where are you coming from, or, rather, which is your closest coast?
The USA is an incredibly diverse country in terms of landscape, climate, and culture. We got mountains, a huge desert (bring lots of water! No, really! There’s a reason that Death Valley is called that), frozen tundra, breathtaking cityscapes, tropical islands with coconut trees where it never snows, quaint farming villages where it’s acceptable to park your tractor downtown, ethnic neighborhoods where you’d swear you were in <country other than the US>, incredibly tacky and materialistic theme parks, and lots more.
Your profile says you are in Greece. Is that where you’re originally from? Maybe you’d like some place that doesn’t look anything like Greece or maybe even Europe?
Eat barbecue.
Well, we certainly have plenty of States for OP to choose from!
If you decide that California is IT, then I suggest Northern California, centering around San Francisco and surrounding areas (everything from, say, Monterey on northward).
With a bit of searching, you can find plenty of threads, right here on SDMB within the last two years, full of suggestions for visiting S. F. and other northern California areas.
One warning - things in the US are a lot further apart than they look on maps. I know several Europeans who thought they could drive across the country in a couple of days.
Are there iconic landmarks you’d like to see?
Yeah, really, the USA is huge if you’re used to Europe.
Here, look at this.
Cool! Europe is freakin’ tiny! No wonder they were in a constant state of war for most of human history.
To third or fourth the point, the US has almost the same land area as Europe and half the population. So its both huge and large parts of it are pretty empty (though the empty bits are often worth visiting for the scenery). So you probably want to narrow it down a little, or at least tell us how you plan on getting around.
I’ll plug starting in New England since a) there’s a good mix of cities and natural beauty, and it tends to have a lot more history then the more westerly parts b) stuff’s relatively close together compared to the rest of the country (though you’ll still need to rent a car), c) the south and west of the country are as hot as balls in the summer, d) when your done in NE you can get a train from Boston to NYC and/or DC. and see the sights in some of the more populated bits of the US without having to drive/find parking in them.
If you visit American city,
You will find it very pretty.
Just two things of which you must beware:
Don’t drink the water and don’t breathe the air.[indent]— Tom Lehrer, Pollution, 1965.[/indent]
just go to southern California and New York City, that’s all that matters in this country. At least, that’s what they tell me.
wow, lyrics from a song written almost 50 years ago. what’s your point?
Just being cute. Is that banned too now?
And you’re suggesting southern California? I grew up there 50 years ago. I know from pollution!
The Wild West (or what’s left of it). The frontier is the traditional American destination of the distressed seeking to leave troubles behind. The modern equivalent is still to line up your car’s hood ornament with the setting sun and keep driving until you run out of road.
You could visit some national parks (which preserve the wilderness to which the restless headed). Or you could re-trace the historic trails they traveled. There are local museums that show what happened before and after civilization caught up with them. Some place or other is usually staging a re-enactment of some great event of pioneer times.
Specific destinations? The Reno/Lake Tahoe/Carson City/Virginia City area of Nevada has a pretty broad assortment of these things in a relatively small area. Yellowstone National Park has thermal features that you won’t see the like of anywhere else.
LOTS of places …
Where are you now?
Find a solid rock n roll bluesy band, not your couples makeup or break up song but a band with a following and a cross continental culture to follow as they tour across the USA. Make friends, see the country, see your favorite band and forget about what’s her face.
chela just back from a GOvt Mule concert… 
Imaginary dream itinerary:
You want wilderness and solitude, and your best bet is California. Death Valley, Yosemite, and Point Reyes. Death Valley to forget and start anew, Yosemite for the sheer wonder, and Point Reyes for relaxation and reflection.