"If you were a Puritan" is a complex idea. Let's unpack it a bit.
To begin with, "Puritan" is a more complicated label than, say, "Roman Catholic." First, on the whole it wasn't something people called themselves. The term was originally pejorative, an insult directed at English Protestants who disagreed with the mainstream Church of England (Anglicanism in England; Episcopalianism in the United States and elsewhere). They fell within the broader category of "Dissenter," which was the general term for English Protestants opposed to the Church of England, ancestors of modern Congregationalists, Methodists and many other denominations now considered part of mainstream Protestantism. Both "Puritan" and "Dissenter" were labels imposed from outside, generally by people who disagreed with the people they were describing.
Second, it's old, arguably out of date when we're talking about Jonathan Edwards. The term "Puritan" is dated to the 1560s (see reference). It was most current in the 1600s, when the English Civil War put England under Dissenting Protestant rule and Puritan communities established colonies in what would become the United States. Edwards wrote and preached Sinners in the Hands of an Angry God in 1741.
So what did the people who actually heard and believed Jonathan Edwards call themselves? How did they see themselves? Answering those questions requires us to look at who and where they were.
The records of Edwards actually preaching Sinners in the Hands of an Angry God refer to him doing so in Enfield, Connecticut (at the time, the town was part of Massachusetts) on July 8, 1741. At that time, New England was in the midst of the First Great Awakening.
The First Great Awakening - depending on who you ask there have been between 2 and 4 "Great Awakenings" in the United States - was a revolution in American Christianity, a major move away from the Church of England and from formal ritual and tradition in general, and toward an approach to Christianity its followers felt was simpler, truer and more grounded in Scripture. It was a grassroots movement with a variety of differing, often contradictory views preached with equal fervor. Who believed what at any given time often varied. What the believers of the Great Awakening had in common was an appetite for intense, often emotional religious experiences. Edwards, both as a theologian and as a famous preacher, was a vital figure in the Awakening.
Edwards himself was a Reformed Congregationalist, a church that followed the teachings of John Calvin. Calvin taught that humans were tainted with total depravity, worthless for our imperfection in comparison with the perfection of God. To quote Edwards's own The Justice of God in the Damnation of Sinners (see reference) "sin against God, being a violation of infinite obligations, must be a crime infinitely heinous, and so deserving of infinite punishment." In that view, humans were by nature damned to Hell. God alone, being perfect, had chosen to spare some humans, called the elect, from damnation.
That, then, was the setting of Sinners in the Hands of an Angry God. The vital difference between a modern reader and an 18th century Protestant sitting in the church in Enfield, listening to the sermon, would have been simple: you already believed what you were hearing. Like Edwards, you may have been a Reformed Congregationalist, taught the doctrines of Sinners in the Hands of an Angry God from childhood. You may have been one of the many converts of the Great Awakening, having heard similar teachings elsewhere and been convinced. You may even, as records show many people were, have been converted by the sermon itself.
Regardless of circumstance, you would have had the fundamental difference from a modern reader that to you, total depravity was real. Hell and the elect were real. Even in an extreme case, say an adherent of a completely different faith who heard or read Edwards's sermon by chance, you would know that your friends and neighbors held that sort of belief.
Sinners in the Hands of an Angry God is an intense, frightening text. But, as noted above, the Protestants of the Great Awakening went to church to have intense, frightening experiences. That's how, as Rev. Steven Williams reports, a church full of people reacted to the sermon with "a great moaning and crying through ye whole House, what Shall I do to be Sav'd—oh, I am going to Hell—oh, what shall I do for Christ., &c., &c..." yet "oh ye cheerfulness and pleasantness of their countenances yt received comfort." That's what church was, and what religion taught, in that place and time. The "Puritans" who heard Jonathan Edwards would have expected no less.