It all depends, really, on what time period you are interested in discussing. As the previous contributors both suggested, American society was largely rural and small-town oriented until both immigration and internal migration produced a society that was largely urban (meaning, by U.S. Census terminology, living in settlements of more than 2,500 inhabitants). This doesn't really happen, definitively, until the late-19th/early-20th century.
In reference to your question about "character" of the population, there are countless ways to define "character." You might think about ethicity, race, and national origin of the population and how it changed over time and had an impact on the nature of American society as a whole. Immigration, of course, has been a constant force in American history and culture since the arrival of the first European settlers in the 16th century, but waves came from different parts of the world and people came for dramatically different reasons throughout the next 400 years.
If you are interested in the history of immigration and the influence of the resulting population shifts on American society and culture, a great book to consider is Ronald Takaki's A Different Mirror.
During the time period in question, the US population increased in size and became more industrialized and urbanized. These changes made the US economy more of a "market economy" and lessened the degree of cohesion in the society.
During this time, the US changed from a society of subsistence farmers who did a bit of bartering with one another to a society in which the economy was based on buying and selling things for cash. This came along with the market revolution and transportation revolution of the early to mid-19th century. This change in the economy (which was tied to the growth in population) made it harder for Americans to feel close to their fellow citizens in the way that people did when they lived in small communities and personally traded with and helped one another
The U.S. will not really have a major population boom until the mid to late 1800's. Prior to that, the U.S. was very rural and economies were localized. The railroad will play a major roll in transforming America as goods and services will now be more accessable to markets across the country. Add with that, the influx of cheap labor provided by immigrants, and the U.S. city as we know it will begin to develop. Mix in other new technologies brought on by the Industrial Revolution, and you have the making of an American economic shift.