Based on personal experience, and the questions posted in this sub, the typical American lacks a meaningful understanding of pretty much every aspect of indigenous history. There are a ton of fascinating books, covering a huge range of topics related to the deep history of the New World, but those books rarely grace the shelves of local bookstores, and rarely find their way into people's hands.

I'm going to recommend some of my favorites below, in a rough order of increasing difficulty for an absolute newbie, and detail why they are my favorites. However, please let me know if there is a specific place/time/people of interest, and I can make more targeted recommendations.

• Charles Mann 1491: New Revelations of the Americas Before Columbus is a great place to start your journey. Mann is a journalist, not a historian, so he oversimplified some complex topics, but he crafted an engaging introduction to the history of the New World. Most newbies cite this book as sparking their love of New World history.

• An Indigenous Peoples History of the United States by Roxanne Dunbar-Ortiz is also a great place to start if you want a grand overview of indigenous history from an indigenous historian. Again, very helpful and engaging for absolute newbies.

• The Heartbeat of Wounded Knee: Native America from 1890 to the Present by David Treuer is another great book from an indigenous historian, and as the title indicates, explores more recent history. Again, a good general introduction if you, like most people, kinda lose the thread of Native American history after 1890.

• Matthew Restall Seven Myths of the Spanish Conquest is a mind-blowing book. He establishes seven persistent myths of the conquest, then breaks those myths down in one brief volume. Forget what you think you know about the early colonial period, and be prepared for a deeper, richer story than you could ever imagine.

• Daniel Richter Facing East from Indian Country: A Native History of Early America is a great introduction to eastern North American history, and like Restall's book above helps to shift your understanding of the narrative of contact away from the European perspective, and instead anchoring the story in Indian Country.

• Andrés Reséndez The Other Slavery: The Uncovered Story of Indian Enslavement in America is the single best introduction to understand the temporal, geographic, and cultural magnitude of the native slave trade in the Spanish Empire. Absolutely vital for understanding the history of the Americas, and almost no one outside of history nerds has heard about the impact of indigenous slavery on the history of the New World.

• Jeffrey Ostler Surviving Genocide: Native Nations and the United States from the American Revolution to Bleeding Kansas is an amazing book that details the violence of early U.S. Indian policy, and the creation of an unhealthy world for Native Americans. Ostler details how Native nations fought for sovereignty in the face of an aggressive, expansive neighbor bent on their removal. This is part one, a forthcoming part two will focus more on the western experience, and I really can't wait.

• Colin Calloway One Vast Winter Count: The Native American West Before Lewis and Clark is the best introduction and overview of the American West. I absolutely adore this book. I recommend it all the time because it blew my mind the first time I read it.

Hope this helps you on your reading journey!

European history on the eve of colonization is usually left out entirely in my experience. It’s especially problematic when trying to learn about a colony like say Plymouth. Without knowing anything about England prior to the Pilgrims departure their story is pretty incomplete. Considering how much emphasis US history places on Plymouth and Massachusetts Bay it’s a little odd (wink wink) that almost no Americans have any idea what the version of England was like that they left behind or even that there was an English Civil War not long after over some of the same issues the Pilgrims and other groups had been griping about decades earlier. These “gaps” also lead to a lot of half truths and misconceptions. It would be fair to argue that the average American’s knowledge of the Plymouth story is more American mythology than history.    

Since I have a lot of interest in family history and genealogy and have personal ties to the Pilgrim/Puritan migrations I’ve spent a lot of time trying to acquaint myself with this period. Some books I would recommend are:

The Life of Elizabeth I by Alison Weir and King James by Pauline Croft. Good intros related to the respective Monarchs that ruled during the Pilgrim’s time of departure.

https://www.goodreads.com/book/show/111222.The_Life_of_Elizabeth_Ihttps://www.goodreads.com/book/show/1194111.King_James?ref=nav_sb_ss_2_12 

Of Plymouth Plantation by William Bradford, Bradford was the second Governor of Plymouth Colony and he has a lot of firsthand info on the early years. 

https://www.goodreads.com/book/show/380887.Of_Plymouth_Plantation_1620_1647?ref=nav_sb_ss_2_8 

Mayflower: A Story of Courage, Community, and War by Nathaniel Philbrick is one of the better books on Plymouth by a modern author. 

https://www.goodreads.com/book/show/4820.Mayflower?ac=1&from_search=true&qid=yQqApxLh1q&rank=1 

The Puritans American Family Portrait by Jack Cavanaugh is historical fiction that is firmly within the Christian lit category, however his story is pretty easy to follow and most of the key aspects of the later Puritan migration are covered. Not a bad way to dip the old toe in the water when learning about a new period of history. 

https://www.goodreads.com/book/show/420224.The_Puritans?ref=nav_sb_ss_1_12 

Albions Seed by David Hackett Fischer’s section on East Anglia to Massachusetts is a pretty great overview of what people’s habits were and what their lives were like.  

https://www.goodreads.com/book/show/32081.Albion_s_Seed?ref=nav_sb_ss_1_12

To begin answering your question, for anything Reconstruction Eric Foner is gonna be your first stop. His A Short History of Reconstruction is very readable and a great place to get a good view of the workings, failures, and successes.

As to the other part, it may not necessarily be entire gaps, though I'd venture based on teaching experience that the French and Indian War, the early Republic through the 1840s, and the period from 1890-1910 are not given as much focus as could and should be.

For me is usually lacking the understanding that its complicated. Laypersons tend to like snippets of factual information without the nuance or context. A meme on Facebook shared by 10 thousand people may be technically true, but may also be severely lacking any depth of understanding. Take for example Thomas Jefferson and Sally Hemings. To know that Jefferson fathered children with her is factual, but there are literally entire books (see Annette Gordon Reed for the essential one) written about how complex the relationship was. To study history is to study nuance, and change over time. Lincoln in 1830 isn't the same Lincoln in 1864, so a quote of his as a young man doesn't encapsulate the man's entire life or thought process, anymore than an older one suggests he always thought that way.

History is very rarely simple and straightforward. People tend to prefer a 5 minute colorful YouTube video than a 400 page carefully researched book. Not that everyone needs to do that, but many people sadly don't want any more information. They got what they came for, so to speak from that video, that short article, that factoid or meme, and don't desire to learn more about how complicated it is.

Foreign policy from 1895-1940, focusing on Latin America.

Most people are unaware of our multiple interventions and nation-crafting there. We created Panama by supporting a rebellion because the Colombians wanted a more favorable deal for the rights to build the Panama Canal.

We invaded Nicaragua twice.

We created Cuba by tossing out the Spanish, in the same war that we created the Philippines.

We shelled Veracruz in 1915.

Book reco: Encyclopedia of U.S. military interventions in Latin America by Alan L. McPherson.

Why is this important? It set the stage for our aggressive regime-change experiments elsewhere in the post-WW2 era that are still happening today.

The War of Jenkins' Ear was one of the British-Spanish colonial wars. It gets it's peculiar name from an event where the Spanish Coast Guard (essentially a band of unregulated but state sponsored privateers) accosted a British ship accused (and probably guilty) of smuggling and chopped off the captain's ear.

The War itself was not exclusively brought about by the Jenkins' Ear incident as that event happened 8 years before the 1739 declaration of war. Rather, tighter Spanish regulation of Anglo-Spanish trade, Spanish manumission of escaped British slaves, and the British establishment of Georgia in what Spain perceived to be the northern half of Florida brought the war to its boiling point.

Although the war ended in a stalemate, it was one of the first truly international conflicts, with fighting happening in North and South America, the Caribbean, Europe, and the Pacific.

American enslaved or recently manumitted populations had a major role in the conflict. Just as news of the war declaration reached Charleston, the massive Stono Slave Rebellion fired off, forcing the British to contend with unrest in their American colonies. Freed Black men in Florida garrisoned Fort Mose just north of St. Augustine. In a bloody battle, they defended the city against a British invasion from Georgia.

Some of my favorite sources on it:

• Carl Swanson, Predators and Prizes: American Privateering and Imperial Warfare, 1739-1748. Covers the naval and privateering aspects of the war, albeit from an Anglocentric perspective.
• Kathleen Deegan, Fort Mose: Colonial America's Black Fortress of Freedom. Talks about Fort Mose, Black flight from Carolina and Georgia to Spanish Florida, and escaped enslaved people's role in the War
• I have others that cover it tangentially, mainly from a maritime perspective as that's what I tend to focus on, but there's just not that many books that exclusively focus on the war.