To what extent did the American Revolution change American society politically, socially, economically?

Socially and economically speaking the Revolution did not have a major impact, indeed those who were part of the ruling classes remained in the upper classes. Slavery was not abolished after the Revolution, though in the North it was abolished shortly after the revolution.

Politically speaking it led to the creation of the Republic with its principles of liberty. The republic was inspired by the ideals of John Locke. The colonists were no longer the subjects of the British crown.

Politically there was a much higher participation of people in politics.
The elite that had been running the colonies now had to gain the support of the common people. Local offices like county sheriff and major now were open to the common people.

Economically the American revolution left the nation as a whole deeply in debt. Trade with England which had been the major trading partner was slow to rebound after the war. The American colonies experienced an economic depression.

However the Middle class experience an explosion. The main means for common people to gain wealth was land. The Northwest Act opened up vast territories of land. A common person could claim a tract of land, homestead on it and the land became his. The American people flooded across the frontier to claim new lands. The line established in 1763 by the British to block new settlements was dissolved.

Socially the elites of the East coast cities kept their positions of wealth. Except those who had supported the British in the war for independence the "Torries" lost everything and many immigrated to England. New avenues for wealth were opened up in shipping, manufacturing and in the south agriculture. American society became as least in theory an open society where the poorest person was not limited to the status he or she was born to.

President Jackson who was born very poor became President, and very wealthy. Davy Crockett born to a western settler family was not only famous but at one time was a United States Congressman. The same laws applied to everyone and there was a least a dream that everyone had the opportunities to gain wealth and social status.