As with any question of this type, there will be different answers because everyone has a different metric of what they deem as "important." I invite you to look at what is here and compare it with your thoughts about the reading.
1. Columbus's own impressions of indigenous people- I think that one of the most important elements of the reading comes from Columbus's own words about his initial interactions with the Arawak:
They ... brought us parrots and balls of cotton and spears and many other things, which they exchanged for the glass beads and hawks' bells. They willingly traded everything they owned... .They have no iron. Their spears are made of cane... . They would make fine servants.... With fifty men we could subjugate them all and make them do whatever we want.
Columbus's intent is very telling. It is not heroic, seeking to expand on the spirit of expansion. Rather, Columbus is driven by the idea that he can "make them do whatever" he wishes. Tearing off the mythology that is so heavily associated with Columbus is one of the most important elements from the first chapter, and as will be seen, history in general for Howard Zinn.
2.Economics drives American History- One of Zinn's fundamental premises in the entire work emerges in the first chapter. Economics is the driving force behind the American narrative. Once again, he challenges the idea that America is exceptional in that it was founded on an idealistic premise. Zinn is direct in suggesting the reason America was founded for economic expansion. Zinn suggests that Spain guaranteed " Columbus 10 percent of the profits" if he found wealth in the new world and fueled Columbus' conquest of this new area. He argues such origins came to define American History, that the drive for property was an organizing principle in the founding of America, as seen in Jamestown, Plymouth, and settlement in Colonial America.
3. Genocide and enslavement is part of Columbus's legacy- Zinn points out that any study of Columbus must conclude that he subjugated the indigenous population who were unaware of his intent until it was too late. Many of these people died under the conditions that Columbus to which they were subjected.
4. Ideology drives historical scholarship- Zinn pivots to talk about the purpose of the historian. One of his most significant points out of the first chapter is how there is no such thing as the completely detached historian:
The historian's distortion is more than technical, it is ideological; it is released into a world of contending interests, where any chosen emphasis supports (whether the historian means to or not) some kind of interest, whether economic or political or racial or national or sexual.
In the end, all history is told from a slant or bias. Rather than denigrate this, Zinn suggests that we must understand this and use it to guide us in how we view history. This moves history from "the truth" to understanding it as "one kind of truth."
5. American History is the collection of many voices and not just one voice- Zinn takes aim at the idea of "the United States" in the first chapter. He suggests that there are many interests in the so- called "United States" and to think otherwise is a way to silence these voices.
6. Kissinger and Zinn- Henry Kissinger suggests that "history is the memory of states." Not surprisingly, Zinn offers his take which is that history should be the narrative of the people. He suggests that he wants to tell history from the standpoint of those who are seen as "the others." The Arawak, the slaves, the workers, those who have been silenced are important parts of who Zinn considers "the people" in his scholarship.
7. Rethinking the term "Indians"- Zinn makes the argument that language is a reflection of power. Indigenous people are often called "Indians" because Columbus made a mistake: "Columbus called them Indians, because he miscalculated the size of the earth. In this book we too call them Indians, with some reluctance, because it happens too often that people are saddled with names given them by their conquerors." Zinn's point is very relevant in that those in the position of power usually tell the story of history in the way they see fit.
8. Progress does not always have to silence those that suffered for it- It is important to note that Zinn is not saying we must forego progress. Advancement is a part of human life. However, he is suggesting that we analyze the cost of such progress and seek to do better. For example, the settlement of the New World is the reason why America comes to being. Zinn is not suggesting that we reject America. Rather, he is saying that part of progress can be self- reflection and examination. There are ways in which advancement can be made without destroying lives in such vast quantity. Zinn writes his book with this call for reexamining "romantic mythology" in American History.