A variety of answers can be given to this broad question. It truly depends on one's own vision of historical significance and personal interests. Personally, I would identify World War II, the civil rights movement, and the Great Depression as three of the most important things one could learn about United States history since 1877.

To begin, World War II impacted the entire world in a way that is still being felt today. National borders, ally agreements, and military spending are just a few of the major changes that WWII brought on. Moreover, it shaped the way the United States was viewed around the globe. The US was able to take on a larger leadership role, starting with the rebuilding of Europe.

The civil rights era is arguably the most important time in American social history. Nearly 100 years after the Civil War, African Americans were not treated as equal under the law. Peaceful protests and several landmark Supreme Court decisions changed the character of the US. With Jim Crow gone, America began to recover from an embarrassing record of discrimination and injustice toward African Americans.

Finally, the Great Depression is yet another important event to study in American history. While the speculative causes of the depression are valuable to study, it is the way the government responded that makes it such a significant event. A nation built on the idea of small governance was totally transformed into nearly a welfare state. However, without these critical moves by Franklin Roosevelt, millions more would have been out of work. The Great Depression still has an impact today, with many programs of the famous alphabet agencies still around.

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.