How did economic changes transform the American class system and the relationship between classes?
I assume that you are talking about the changes early in the history of the United States -- changes that went along with the very early stages of industrialization.
In general, historians tend to argue that this was a time during which there came to be more of a distinction between classes and more tension between the classes. Industrialization tended to create more inequality, just as the Jeffersonians had feared that it would. Instead of having a relatively homogeneous population of small farmers, the Northern US came to have a differentiated population with a large working class of people who had relatively little money.
You can argue that you start to see evidence of class tensions in things like the election and presidency of Andrew Jackson. You can argue that his Bank War is an example of how the poorer people distrusted the rich and felt resentment towards them.
It depends on the time frame of the changes you are referring to. For more than a century after the beginning of the United States, there was virtually no middle class. A wealthy elite ran the major levers of the economy and most of government with it, while the vast majority of Americans - in excess of 90% - lived in relative poverty. When there is that disparity of wealth, there is class conflict, as evidenced by the organized labor and populist movements of the late 1800s.
Changes beginning in the 1920s to the way we paid our workers, the way work was done and the size of our industries created a mostly permanent middle class that only grew over time. This reduced class conflict, although the relationship between the middle and upper classes is still a bit contentious these days.