This question is fairly subjective so I'll answer it with three of the most significant events in US history from 1877 onward. You may also want to read an overview of US history from 1877 onward to learn about events with personal significance to you.
1) The Great Depression
After World War I, there was a time of economic growth and freer culture in the United States, which is what made the Great Depression so profound and unexpected. The life of excess that many US citizens lived came to an abrupt and painful halt, with banks, investments, and savings accounts all collapsing. In the height of the greatest economic depression in US history, over 15 million Americans (or a quarter of the population) were unemployed. The Great Depression not only shaped American culture up to and during World War II, but also brought about welfare programs, like Social Security, that are still in effect today.
2) World Wars I and II
While the World Wars' impacts stretched beyond US history, they influenced modern American culture in powerful ways. These were the deadliest wars in human history, with lethal weapons used that never had been before and significant emotional trauma on those who survived. The bombing of Pearl Harbor, in particular, was a turning point in US history because it engaged the US personally in global conflict and eventually contributed to the weaponization of nuclear power.
3) Space Exploration in the 1960s
Thanks to the "space race" between the United States and the Soviet Union, US astronauts landing on the moon became not only one of the most significant events in US history but one of the greatest scientific accomplishments. The 1969 moon landing expanded human exploration to the stars and brought a sense of hope and peace to a troubled, politically tense time in American history.