tl;dr
Based on his family background, the grammar school curriculum of the Elizabethan era, and some scenes in his plays, we can conclude that Shakespeare had training in classical rhetoric.
Did Shakespeare attend school?
There are no documentary records that Shakespeare attended school at all. However, his family background makes it very improbable that he would have had no formal education. William's father John Shakespeare was a respectable citizen of Stratford. In 1568 he was elected Town Bailiff, which corresponds roughly to mayor. Shakespeare would have been four years old at the time. Children typically began their schooling at age four or five at a "petty school," where they learned to form their letters and to read English. After two years, they went on to grammar school. The local grammar school's charter specified that it was a "Liberam Scolam Grammaticalem," i.e., a free school where students paid no fees (Baldwin Vol. I p. 464). So while a poor family might need the child's labor or might not be able to afford school materials, children from middle-class families would attend school as a matter of course. Samuel Schoenbaum sums up the situation:
The parchment registers for Stratford pupils of this period have also, not surprisingly, perished. More than a century later in 1709, Nicholas Rowe remarked that John Shakespeare had bred his poet son "for some time at a Free-School, where 'tis possible he acquir'd that little Latin he was Master of". This is our earliest allusion to Shakespeare's formal education, and may either record a tradition or reflect Rowe's natural inference. If an inference, it seems valid enough: we need not doubt that Shakespeare received a grammar-school education, and the only likely place for it was the King's New School of Stratford-upon-Avon. The schoolroom stood in Church Street, behind the Gild Chapel, and only a quarter-mile distant from the family house in Henley Street. (pp. 62–63)
Shakespeare is unlikely to have attended university. Beginning around 1576, John Shakespeare fell into financial difficulties that lasted about ten years. Since schoolboys typically attended grammar school until age fourteen, Shakespeare would have been eligible to enter university around 1578, but it would not have been affordable for his family at that time. So his education would have been confined to the grammar school curriculum.
Grammar school curricula in Elizabethan England
What would that curriculum have been? While not entirely standardized, it was nevertheless quite similar across schools. Schoenbaum says:
The curriculum of Stratford grammar school has not come down, but we can reconstruct it with some confidence on the basis of what we know of similar foundations in other parts of England. (p. 62)
T. H. Baldwin provides further details:
[B]ecause of the general uniformity of curricula and methods at the time, the particular school in which Shakespeare received his training is not for our present purpose of great importance. It was the conscious purpose of the authorities so to regiment schools and texts that transfers in teachers and schools would not retard the progress of the pupils. Because of this regimentation, the curricula had a certain common denominator of uniformity. (Vol. I p. 557)
The curriculum consisted more or less exclusively of training in Latin. After spending the first couple of forms learning the rudiments of grammar and vocabulary, students went on to read classical authors. Baldwin quotes statutes for a school in York, too long to be reproduced in their entirety here; however, here is a sample:
In the third forme shalbe placed all Schollars that have profitablie passed through and learned ye premisses before prescribed And the Mr shall teach to them the Latin Grammar as it is set forth & used in this realme. Terence Esopes fables virgill Tullies epistles or so many of them as he shall thinke fit for the capacitie & profit of his schollars in the same & as he shall percieue them profit in Learning so hall he place them in the 4 forme——
And euery day he shall giue them an English to be made in Latine & teach unto them there placed Salust, Ovid, Tullies offices, the commentaries of Caesar, Copia verborum et Rerum Erasmi; or so many of the said Bookes or or other as he shall thinke expedient for their Capacities and also he shall teach the art & rules of versifieing (if he himself be experte therein)... (Vol. I p. 431)
The standard textbook was Lily's Latin Grammar. In 1542, Henry VIII proclaimed that this two-part book would be the only authorized and obligatory text that schoolteachers could use (Lily vii). The first volume, Introduction of the Eyght Partes of Speche, was in English, and provided basic grammar drills, with sample sentences from classical authors. The second volume, Brevissima institutio ("Brief Introduction"), used after the first couple of years, was in Latin. It provided lessons in advanced grammar, composition, versification, and rhetoric. Students were also encouraged to read Latin authors in a specific, four-fold way. Peter Mack explains the approach:
In his instructions for reading, which were included in Lily's Brevissima institutio, Erasmus suggested that pupils should re-read texts four times: at first straight through to record the general meaning more thoroughly; then word by word noticing vocabulary and constructions; thirdly for rhetoric, picking out figures, elegant expressions, sententiae, proverbs, histories, fables, and comparisons; and finally ethically, noting exemplary stories and moral teachings. (pp. 14–15)
Once the students had learned the rudiments, then, they would be reading actual Latin authors, not just in order to translate, but also to study rhetoric. An orator whose works were an integral part of this curriculum was "Tully," i.e., Marcus Tullius Cicero. Students would be expected not only to read and translate Cicero's speeches, but also to render English passages into Ciceronian Latin. An study of rhetoric would be inextricably linked to these exercises.
Some grammar schools did teach Greek to students in the highest forms. However, Greek did not form part of the curriculum at every school, and it is unclear whether Shakespeare would have studied any Greek texts on rhetoric at the King's New School.
Evidence for Shakespeare's Schooling in the Plays
Baldwin thoroughly documents instances where Shakespeare's plays show evidence of his having undergone the standard grammar school curriculum. A famous scene from The Merry Wives of Windsor shows a young schoolboy named William being drilled in elementary Latin:
SIR HUGH That is a good William. What is he, William,
that does lend articles?
WILLIAM Articles are borrowed of the pronoun and be
thus declined: singulariter, nominativo, hic, haec,
hoc.
SIR HUGH Nominativo, hig, haeg, hog. Pray you, mark:
genitivo, huius. Well, what is your accusative case?
WILLIAM Accusativo, hinc.
SIR HUGH I pray you, have your remembrance, child. Accusativo, hung, hang, hog. (IV.i.38–47)
Baldwin gives several such instances where Shakespeare's debt to the first volume of Lily is clear: not only in drills, but also in quotations of tags included therein.
Baldwin also provides instances of Shakespeare's use of the second volume of Lily. One famous example is the quotation from Horace:
Integer vitae, scelerisque purus,
Non eget Mauri iaculis, nec arcu (Odes, I.xxii.1-2)
Translation:
The man clean of life and pure of wickedness
Needs neither the arrows nor the bow of the Moor.
In Titus Andronicus, Titus sends his daughter's rapists this note, along with some weapons. On reading it, Chiron, one of the rapists, remarks:
Oh, 'tis a verse in Horace, I know it well:
I read it in the grammar long ago. (IV.ii.22–23)
This quotation is indeed in Brevissima institutio. As Pramit Chaudhuri points out (pp. 789-790), Chiron and his brother Demetrius recognize the quotation, but fail to understand its meaning in this context. Shakespeare takes a standard tag from the Latin textbook, but uses it to great dramatic effect. Sending weapons to the rapists while claiming that upright men have no need of them simultaneously positions Titus as an upright man, since he is giving away his weapons, and marks the brothers as villains.
In addition to such direct quotations from the grammar, Shakespeare's plays also show indirect evidence of training in rhetoric. Krystyna Kujawińska-Courtney says, for example, that the playwright "very often appropriated classical form without necessarily borrowing the content itself" (p. 214). She says Julius Caesar shows Shakespeare using the techniques of classical rhetoric without any direct borrowing:
Shakespeare extrapolated rhetorical questions and ad hominem argument—two formulas of persuasive oratory from the works of Quintilian and Cicero. When he found no exact source in Plutarch for the orations of Brutus and Antony in Julius Caesar he embodied what he made Brutus and Antony say in these two formulas. Actually, Julius Caesar is full of other persuasive speeches: Marullus haranguing the Plebeians in the street; Cassius wooing Brutus towards conspiracy; later, Cassius doing the same to Casca; Portia pleading with Brutus in their orchard, and so on. These speeches, too, made use of the two rhetorical devices Shakespeare thought of as the orator’s stock in trade. (p. 215)
Baldwin too provides examples of Shakespeare's using rhetorical forms to structure dialog. For example, the Rhetorica ad Herennium, attributed to Cicero, discusses a way of conducting a judicial inquiry by conjecture: the examining judge entertains various conjectures as to what might have happened before hitting upon the truth by coming up with the one conjecture that will fit all the facts. Baldwin says that the trial scene in Romeo and Juliet (V.3), where Friar Laurence is suspected of murder before the facts come to light, follows the pattern of conjectural investigation laid out in the ad Herennium (Baldwin Vol. II p. 76–81). Baldwin's second volume contains several such instances of Shakespeare's plays using rhetorical techniques from various classical texts that would have been part of the grammar school curriculum.
Conclusion
While there is no direct evidence that Shakespeare attended school at all, the circumstantial evidence of his having done so is overwhelming. A boy of his family background would have been sent to the local grammar school. Similarly, while there is no direct evidence of the curriculum at the Stratford school, the standardization of education that began with Henry VIII and prevailed throughout Shakespeare's schooldays and beyond provides sufficient grounds for confidence regarding what he would have learned. He would have had a thorough grounding in Latin grammar and rhetoric, and would have been expected to both identify rhetorical techniques, and use them in composition. His plays provide ample evidence of the use of such techniques. From this, we may safely conclude that Shakespeare had training in classical rhetoric.
References
Baldwin, T. W. William Shakespeare's Small Latine and Lesse Greeke. 2 vols. Urbana: U of Illinois P, 1944. On Archive.org: Volume I, Volume II. Accessed 25 Jan 2021.
Chaudhuri, Pramit. "Classical Quotation in Titus Andronicus." ELH, vol. 81 no. 3, Fall 2014: pp. 787–810.
Horace. Q. Horati Flacci Carminum Liber Primus (Odes, Book I). Accessed 25 January 2021.
Kujawińska-Courtney, Krystyna. "Controversies over Shakespeare's Classical Education". Collectanea Philologica 3 (1999): 207–215. Accessed 25 January 2021.
Lily, William. Lily's Grammar of Latin in English: An Introduction of the Eyght Partes of Speche, and the Construction of the Same. 1542. Ed. and Intro. Hedwig Gwosdek. Oxford: OUP, 2013.
Mack, Peter. Elizabethan Rhetoric: Theory and Practice. Cambridge: Cambridge UP, 2002.
Schoenbaum, S. William Shakespeare: A Compact Documentary Life. Abridged ed. of William Shakespeare: A Documentary Life (1975). New York: Oxford, 1977. Accessed 18 Jan 2021.
Shakespeare, William. The Merry Wives of Windsor. 1597. The Folger Shakespeare. Accessed 25 Jan 2021.
———. Titus Andronicus. 1594. The Folger Shakespeare. Accessed 25 Jan 2021.