Shakespeare's plays (and much "high art" in general, operas, symphonies, etc) held a much different status in 18th and most of the 19th centuries. Today, we think of Shakespeare as something that must be appreciated on an intellectual level, the province of the erudite upper class. In 1790, Shakespeare would have been much more ubiquitous along the social strata. Farming communities would host their own performances of Hamlet and King Lear, lively with singing, dancing, and heckling. On-stage performances with quiet, respectful, and wealthy audiences enjoying the show would have been an alien scene to most Americans prior to the later 19th century.

I cannot speak to how a schoolchild would learn Shakespeare in the classroom in 1790 (I would imagine not at all, except perhaps in the most elite classes). It would not be read by most Americans, as few were literate or educated on such a level. However, most people, rich or poor, urban or agrarian, would be broadly familiar with Shakespeare's works as pop culture. In these slapdash community performances, themes, motifs, motivations, and dialogue especially would differ greatly. It was not until the Gilded Age that this popular public ownership of Shakespeare would change. This is largely, in my estimation, owing to the increased social stratification in urban areas at this time, and the construction of great theaters and music halls, wherein the codification of etiquette was used as a means of social control and social exclusion.

Sources: Highbrow/Lowbrow: The Emergence of a Cultural Hierarchy in America. Levine, Lawrence (1988).

The Incorporation of America: Culture and Society in the Gilded Age. Trachtenberg, Alan (1982).

I have an answer to an earlier question that I think will answer yours more directly:

• When did Shakespeare become "the best playwright ever"?

Shakespeare's reputation has certainly come a long way from Robert Greene's cranky 1592 reference to "an upstart crow" who thinks that just because he can act, he can write plays, too. But while Shakespeare won over contemporary audiences and other prominent writers, it was a long and winding path from "writer of regionally popular plays" to "veritable god of an international dramatic and literary tradition."

Most basically, in Shakespeare's day the idea of playwright as credible public persona/career was just barely in its infancy, maybe still in utero. Printed copies of individual plays were marked based on the company or even the actors that had performed them, not their authors. Nevertheless, shortly after Shakespeare's death, several colleagues published a collected edition of (many of) his plays. Their stated purpose was to correct the errors in copies of his plays circulating individually. In other words, by 1623, there existed both the idea that a playwright could have a culturally significant legacy as a playwright--and that Shakespeare had left a legacy necessary to protect.

But the course of that legacy for the first century or so was already set in Shakespeare's lifetime. Effusive praise poured forth: Shakespeare was the equal of the classic English writers: Chaucer, Gower, Spenser; no, Shakespeare was greater. But! But. Shakespeare possessed an innate, raw, rustic talent; he lacked education and refinement. He was the playwright for the people, not for the literati. His contemporary Ben Jonson, who outshone Shakespeare in reputation through the 17th century, remarked that Shakespeare's work "wanted art" (lacked). He /r/badhistory-ed the other's plays, pointing out errors in history and geography that any learned man should have known.

Shuttering the theaters during the Interregnum caused another leap in status for plays and their authors, as literary critics began to view plays as the stuff for critical analysis, not just performance. Shakespeare's early reputation as raw natural talent fared poorly. Critics pointed out all the ways his plays--even, or especially, the vaunted Hamlet--failed to measure up to the dominant neoclassical dramatic theory. (In short: nothing about the ghost makes sense)...all the while praising his innate ability to become his characters, to make fiction seem real. We revere this as art today; not so the leading literati of the late 1600s.

But the roots of the shift were already being laid. Michael Dobson has traced how later authors' adaptations of Shakespeare (in the sense of rewriting, not "Macbeth in a Pennsylvania restaurant") were already working to smooth out perceived rough edges and bring his works more in line with ideas of neoclassical refinement. This is just one example of how Shakespeare's plays served as almost a canvas for different groups in Restoration England to project their sense of identity or politics onto. Above all, a mainstream emphasis on seeing "domestic virtue and triumphant colonial warfare" in his plays gradually led to Shakespeare's crowning as the national playwright-poet of England.

Shakespeare didn't attain the nickname "The Bard" until the 19th century and George Bernard Shaw would not coin the term "Bardolatry" until the next. But by 1769, actor and theater bigwig David Garrick could organize a "Stratford Jubilee" celebrating Shakespeare, and unsarcastically perform an ode proclaiming: "Tis he, tis he! The god of our idolatry."

So Shakespeare's enshrinement as the preeminent literary voice of England was secured in the first half of the 18th century. Up until the point, he had enjoyed basically zero international reception. Voltaire, in his obsession with English post-Restoration culture, is usually considered to have "introduced" Shakespeare to the wider European literary elite. But his 1733 assessment was ambivalent:

a fecund genius, full of vigor, ranging from simple naturalness to the sublime, without the least glimmer of taste or the slightest knowledge of the rules.

Indeed, Voltaire was toeing the 1600s English party line, the neoclassicizing Age of Reason. It was not until the nascence of romanticism in late 18th century Germany that critics and authors would claim Shakespeare as the paragon of a literary world that transcended national borders. To Gotthold Ephraim Lessing, the much maligned ghost of Hamlet is a true character that made audiences believe in the enchanted world again, even in an age of reason. Johann Wolfgang von Goethe insists that if Shakespeare's dramatic logic appears illogical, the fault is in us, not the play: we need to insert ourselves into his world. Works of art make their own rules; we need not apply external ones.

These German proto-romantics' use of an English author, instead of a native one, as their exemplar by the turn of 1800 completes Shakespeare's rise from upstart crow to rustic talent to English paragon to international superstar.

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Sources: Michael Dobson, The Making of the National Poet, is the classic investigation of how Shakespeare's reputation evolved in the first century and a half.