I'm Danish, so not sure if my answer counts, but I know a few American expats here. To be honest, some hate it here, and I mean hate it. They don't like the culture, they don't like the language, a few have even given up ever speaking it, they don't like the people. The only reason they stay seems to be for the sake of their Danish husband or wife.
Others, while they still have language and culture issues, especially at the beginning, love it.
Much of how well you settle in a new country will depend on you as an individual. From what I can see, the difference between the expats that settle well here and those that don't comes partly from their attitude and partly from their culture. Aussies in general, for example, tend to settle much better than Americans.
My advice is to take a long hard look at yourself first, can you laugh at yourself when you make a mistake? Are you okay with speaking baby-language and maybe not be understood at all until you have become more fluent in Danish? Are you okay with people switching to English when they hear your Danish is not perfect? Can you deal with the stress of things not working the same way as at home? Are you outgoing and willing to take the first step in meeting new people? Are you okay with people laughing if you unintentionally say something funny? How do you feel about being sick and feeling miserable and being an ocean away from your family? These sort of things.
Also read up as much as you can about the cultural differences between the U.S. and Denmark. While most Danes speak English and we are both western countries, the cultural differences can be huge, and many of the issues I see actually stem from cultural differences where expats feel they've been treated rudely, but where what they experienced would not considered rude here, and the offender would probably have been absolutely mortified if he knew that the expat had taken offense.
I've been an expat myself, and I've often put my foot in my mouth, and on the other hand often gotten very upset with a behaviour where it was later explained to me that while it was rude, then it was not nearly as rude as I thought it was. I could have been ready for someone to be fired, whereas my local friends were all "naaah, that's normal, happens all the time. Everyone does it.". ;)
I don't have any experience with the Rosetta Stone program, but I will say that Danish is not a language you can learn without hearing it spoken by a Danish speaker and having a Danish speaker there to correct your pronounciation. While Danish grammar is pretty straight forward the pronounciation is considered one of the most difficult in the world and also words and sentences are not always pronounced as they are written.
As a U.S. citizen, you will, however, have to attend obligatory, free, Danish lessons once you've received your stay permit.
******
Rugratzzz: I think that's a very unfair statement. Yes, I am sure that there are some Danes who think Denmark is perfect, although I have never met any. And there is certainly a large group of Danes, probably even the majority, who think that while Denmark is far from perfect, then it is better than anywhere else in the world, except for maybe Norway.
But trust me, many Danes see plenty wrong about Denmark. For example, the current government and its immigration policies is a huge issue, and for me the elections this year cannot come fast enough.
Also, as mentioned, having lived abroad for many years, I can see how the Danish culture can be an issue. It even sometimes makes me cringe when I have Danish friends over when friends or family from abroad are visiting. I never noticed it before, but I certainly do now.
Unfortunately some expats discredit everything come from a Dane whether or not they agree with them, I have long stopped engaging in any kind of dialogue with this particular group (NOT saying you belong to them Rugratzzz, in fact I don't think you do), not because we did not agree on most political issues, in fact we did, but because of the verbal abuse I automatically became the target of based solely on my nationality. I could either "take it on the chin" as they so kindly suggested, and accept being called almost every name in the book, and preferably admitting to them, or I could confirm yet another stereo-type of the Danes being reserved and impossible to make friends with. I choose the latter.
Personally, I like to think of people as individuals and judge them as such, regardless of nationality.
For an expat moving to Denmark, I would obviously suggest not to solely join the "all Danes are pigs" group, nor the group wearing rose-tinted glasses, but to mainly find expats like reminisence80 who seems to have a more nuanced and realistic view of what it is like here - the good and the bad.