There are actually a lot of different questions going on here:

When did people begin to study Shakespeare? When did the study of Shakespeare make its way into schools? When did people start thinking about Shakespeare as a genius?

Interestingly, Shakespeare's fans were trying to sell people on his genius within a few years of his death. In the first collected edition of his works (the First Folio, 1623) his friend and occasional rival Ben Jonson claims him to have been "...not of an age but for all time!". The Globe was pulled down, not because people didn't respect Shakespeare, but because Early Modern London wasn't precious about buildings - the Globe had originally been "The Theatre" but had been pulled down and rebuilt on a new site (with the same materials) in 1599, and was destroyed by fire in 1613. The rebuilt Globe was shut down at the beginning of the English Civil War in 1642 by Puritan lawmakers who objected to theatre on philosophical and moral grounds. It was pulled down in 1644 to make way for homes.

When theatres reopened in 1660, Sir William Davenant led a revival of Shakespeare almost immediately (Davenant liked to imply he was Shakespeare's illegitimate son) a wrote several adaptations of his plays for the tastes of the new audience. In this period we start to see a split between the theatrical use of Shakespeare - re-framing the best bits into crowd-pleasing new plays (and operas) - and the "serious" study of Shakespeare as a poet in printed form. John Dryden, Poet Laureate 1668-88, exemplifies this - quite a tough critic of Shakespeare as a poet while using him as inspiration to write his own plays (such as his Antony and Cleopatra drama, All for Love). All the subsequent Poets Laureate in this period are adapters of Shakespeare to some degree, and finally in 1709 Nicholas Rowe produces what we might call the first edited edition of the plays - Rowe divides the acts into scenes, for instance, and indicates entrances and exits. Over the course of the 18th century there are several competing editions of "readable" Shakespeare, which are intended to be studied rather than performed. This division (between theatre and study) persists well into the 19th century - it's a commonplace to say that when Keats wrote "On Sitting Down To Read King Lear Once Again" he was thinking of reading a (spiky) dramatic poem on the page and not seeing a play - as most *theatre* audiences knew Nahum Tate's adaptation instead (it has a different ending!).

When do people begin to teach Shakespeare in schools? This is a little harder to track - it's worth saying that the UK doesn't have a unified school curriculum until the *1990s* (though Shakespeare is on it), so hunting down the first school to teach Shakespeare might be quite time-consuming. What we can ask is, "when does knowledge of Shakespeare become a requirement?" and that's a bit easier to gauge. The 1854 Macaulay report sets out a series of recommendations for assessing candidates for the Indian Civil Service (administrators for the East India Company); a knowledge of Shakespeare is assumed in the report, alongside the need to be able to translate Latin and/or Greek. This suggests that elite schools had made use of Shakespeare for a few years at this point, and the ones that didn't would have to start if they wanted boys to go on to Indian Service jobs. From this time up to the mid-20th century, the division between stage and study narrows then widens again. There's a revival of "authentic" Shakespeare, ditching the adaptations and using lavish period costumes. There's a tercentenery festival in 1864 and the Memorial Theatre is built in 1879. Study, however, crystalizes at the beginning of the 20th century as the "right" way to study Shakespeare becomes the use of character to explain universal truths about the human condition - a method exemplified by A C Bradley's Shakespearean Tragedy (1904) - and it's this that's taught for exams.

The post-WW2 British theatre boom (and the rise of cinema) blows all this up and since then, performance and study have grown closer together and inform each other. In 1988, the National Curriculum was introduced in England and Wales and Shakespeare was on it (it was a rolling program so many things weren't fixed until the mid-90s) and since then, Shakespeare has been a feature of public exams. I hope that was a comprehensive answer.