Great question! I think you're on the right track: understanding the legacy of the American Revolution is particularly important to understanding the assumptions we make about who counts as an American and who this country works for. There is actually a growing canon of works that study this topic through the lens of the war itself, rather than just focusing on the political aspects of the Revolution. Still, it's good to have a broad foundation. With that in mind, let's build you a bookshelf.
Starting off
Robert Middlekauff, The Glorious Cause - The Oxford History of the United States' entry for the American Revolution, this will cover all the bases, though the book is now old enough to run for president.
James Kirby Martin and Edward Lender, A Respectable Army - A handy military survey of the Revolution that also traces the growth of professionalism and military character in the Continental Army.
Origins of the American Revolution
Why did this whole mess start? What caused tens of thousands of Americans to willingly create new governments and unify across colonial lines in ways thought impossible just a decade earlier?
Bernard Bailyn, The Ideological Origins of the American Revolution - Bailyn, arguably THE Godfather of colonial American history, took a deep dive into the sermons and political pamphlets read by literate, politically-minded Americans - the kind of people who went on to become Founding Fathers. In these, he found deep anxiety about the corrupting nature of power, as well as a veneration of the kind of community-focused, virtuous yeoman farmer that many New Englanders saw themselves as.
Edmund S. Morgan, Inventing the People - Morgan finds that American political thought in years of the imperial crisis (the years between the Seven Years' War and the Revolution) was shaped by an emphasis on popular sovereignty, the concept at the core of no taxation without representation. He traced the origins of popular sovereignty's central place in the American mind through Americans' reading of English history, particularly since the upheavals of the English Civil War and the Glorious Revolution both showed that power rested well outside the person of the king.
Jack P. Greene, The Constitutional Origins of the American Revolution - Greene argues that the literate, politically-minded Americans were most offended by the ways that George III and his Parliament seemed to threaten or break the sacred, long-established norms of the British constitution. This made Americans the true inheritors and defenders of British liberty.
Gary Nash, Urban Crucible - Hey, enough with the literate, politically-minded American elites already. Those stiffs made up a fraction of the population anyway. Instead, Nash sought a bottom-up understanding of why the Revolution came, and found it in the port cities of colonial America. After all, in every colony, the sharpest and most important anti-tax protests took place in the cities. There was a sharp economic downturn after the Seven Years' War, since there were now no armies to feed or transport and men who had been employed by the military now found themselves out of work. Urban dockworkers found themselves caught on both horns of this problem: there was less shipping, and more competition for unskilled labor thanks to the influx of ex-soldiers. Hungry and poor people tend to be angry people, and the growing urban poor found targets for their anger both in the economic elite - the small number of people in each colony that found themselves growing richer and more powerful in the crisis years - and imperial officials who seemed intent on choking off the jobs that came with smuggling.
Alan Taylor, The Marketplace of Revolution - Taylor looks at the broadest base possible in colonial American society: consumers. By the crisis years, the American colonies' primary function in the British empire was as a market for British goods, and Americans defined their place in society by the conspicuous consumption of those goods. The ideal American gentleman lived in a proper English-style house, with English-made furniture, ate off of Queensware plates, and dressed in the latest London styles. Even poor Americans would spend money on a bit of silk ribbon or lace to try to fancy up their clothing and edge out a bit of social advantage over their peers. To people with this mindset, Parliament's restrictions on trade endangered American's very sense of who they were. In response, they organized an economic response in the form of non-importation (turns out "boycott" is a 19th century word!) movements. The organizations that created and supervised non-importation pacts grew into the Revolutionary governments, which themselves marketed and sold the Revolution almost like it was another product.
War and Society
Now we're hitting the good stuff.
Charles Royster, A Revolutionary People at War - Still one of the go-to texts about the Continental Army, even though it came out during the Bicentennial. Royster traces the American people's involvement with the war effort during the Revolution. Things started sunnily enough,with the rage militaire after Lexington and Concord sparking a massive number of American men to enlist in their local militias or the Continental Army. After the campaigns in New York and New Jersey in 1776 showed that war is, in fact, hard and dangerous, many of those who could find something else to do left the service. The remaining soldiers were younger and poorer than free American society at large, and had to find their own way of rationalizing their continued service to and defense of an increasingly detached civilian population.
Judith Van Buskirk, Standing in Their Own Light - This book was only released a few month ago, but looks well on its way to becoming an instant classic. Van Buskirk looks at what the war meant to African Americans, particularly those that fought on the American side. Since the British guaranteed freedom in exchange for service early in the war, why did so many black men choose to fight for a Revolution led by slaveowners (note that FAR more black people took the British up on their offer)? What was their experience like in an integrated army?
Gregory Knouff, The Soldier's Revolution - Knouff examined the social meanings of military service among Pennsylvanians during the Revolution, with a special focus on the importance of community and localism. Serving in the militia was the surest way of proving oneself a virtuous member of the community, and the universality* of militia service united both the rich - now officers -and the poor enlisted men. Why the asterisk, you ask? Because militia service was not ACTUALLY universal; only white men could serve. The ideal of white men coming together to define and defend their community overshadowed the roles of African-Americans during the war and fed into expressly white conception of American society that formed in the nineteenth century.
Carolyn Cox, A Proper Sense of Honor - Cox examined the officers of the Continental Army, and generally found them to be fussy wannabe aristocrats who were very sensitive about honor, reputation, and social position.
Charles Neimeyer, America Goes to War: A Social History of the Continental Army - A more current and focused look who joined the Army and why, with a focus on the enlisted men.
Robert A. Gross, The Minutemen and Their World - A microhistorical look at the communities of Lexington and Concord, which explains militia activity in those towns along social lines.
Holly Mayer, Belonging to the Army - Mayer explores the concept of the "Continental Community" - that the army was, in fact, a mobile society, composed of soldiers, thousands of their wives and children, merchants, craftsman, prostitutes, and others.
Steven Rosswurm, Arms, Country, and Class: The Philadelphia Militia and the “Lower-Sort” during the American Revolution - While the American Revolution always contained elements of a broader social revolution, Philadelphia probably came closest to undergoing a complete social upheaval. The Philadelphia militia, composed mostly of the urban poor, enforced a radical new constitution full of crazy ideas like universal male suffrage and removing property qualifications from office-holding. Eventually, the lower sort proved so violent and disruptive that middling Philadelphians sided with the remaining elites to disband the city militia and institute a more moderate constitution.
Michael McDonnell, The Politics of War: Race, Class, and Conflict in Revolutionary Virginia - McDonnell examined the social conditions in Virginia during the War of American Independence. Unlike the more homogenous population of Massachusetts, Virginians were deeply divided by class and race. While most Virginians were opposed to Parliament’s efforts to tax the colonies, small farmers and poor whites were largely unwilling to take up arms to defend what they largely saw as a planters’ rebellion. At the same time, enslaved blacks took the opportunity caused by social and political turmoil to try to achieve their own freedom, either by rebellion or running away to British lines. These various tensions led to rebellion and civil war within Virginia by 1780. The difficulty of fostering inter-class cooperation during the Revolution caused Virginia to create a more restrictive political system in the years after the Revolution.
To start off, I'd suggest 1776 by David McCullough. McCullough has a gift for storytelling and his works can be easily read by any layperson. If you like his writing style, he has several biographies on several other figures in history.
The Creation of the American Republic, 1776-1787 by Gordon S. Wood. This is another overview and it'll give you a pretty good look at the war.
Then there's The Creation of the American Republic, 1776-1787 also by Gordon S. Wood. This is a bit more scholarly and looks into more of the ideas and politics so you might want to get some ground work before diving straight into it.
There is also our own subreddit book list for you to peruse at your own leisure.
And finally, assuming that you're in school, I recommend talking to one of your teachers/professors at your school/university. Odds are they know you better and can help you find out what will interest you best. Plus it helps to have somebody to ask questions to in person and to help guide you along in your studies, even if they're extracurricular. Most teachers love a student who just wants to learn and, even if they personally cannot do it, point you in the right direction.
Best of luck in your studies and enjoy!