The Two Gentlemen of Verona is very early Shakespeare. It may be his earliest comedy of all, or at least the earliest to survive (though The Comedy of Errors is another contender for this title). It has a feel of youthfulness – the two eponymous gentlemen and their tangled love affairs make me think of those "golden lads and girls" in the famous song from Cymbeline. The play is sometimes criticised for its "immaturity", to which the standard rejoinder is that it works much better on stage than on page – a proposition that will be put to the test on 12 July, when Simon Godwin's new production opens at Stratford. It is the first to appear on the main stage there since Robin Phillips's 1970 production with Helen Mirren as Julia and Ian Richardson as a raffish Proteus.

The date of composition is a matter of guesswork, as the play is entirely undocumented: no records of performance, no early quarto edition. The earliest definite notice of it appears unhelpfully late, when it is mentioned in Francis Meres's Palladis Tamia, an invaluable repository of Elizabethan literary gossip published in 1598. Shakespeare is "most excellent in" writing comedies, Meres notes, "witnes his Gentlemen of Verona, his Errors, his Love labors lost, his Love labours wonne, his Midsummers night dreame & his Merchant of Venice". Possibly the list is chronological: much of it (leaving aside the enigmatic "Love labours wonne") accords with our idea of those plays' order of composition.

A broad consensus would date the play to c1590-92, though some would push it further back into the hinterland of the late 1580s – the so-called "lost years" of Shakespeare biography. In this early stratum of his literary activity he was attracting attention as a writer of historical drama (the three Henry VI plays) and tragedy (the blood-soaked Titus Andronicus). The first actual record of a Shakespeare play on stage is a performance of "harey the vi" – almost certainly the play now known as Henry VI Part One – at the Rose theatre in Southwark on 3 March 1592. The same play is praised by Thomas Nashe in his pamphlet Pierce Penniless, published the following September. Robert Greene's Groatsworth of Wit, with its infamous attack on "Shake-scene", was published posthumously in the same month. It refers to him as "an upstart crow … that with his Tygers hart wrapt in a Players hyde, supposes he is as well able to bombast out a blanke verse as the best of you": the italicised words are a direct parody (with "player" substituted for "woman") of a line from Henry VI Part Three.

This surge of publicity around Shakespeare's history plays – partly admiring and partly hostile – has been thoroughly picked over by literary historians. But is there anything about his early comedies? I believe there is. It is to be found in the pages of another pamphlet: Kind-Heart's Dream by Henry Chettle, the portly printer and scribbler who had edited the Groatsworth. He praises Shakespeare's "facetious grace in writing". The word "facetious" (Latin facetus) did not have the overtone of waggish jocularity we now give it: it referred to witty, elegant, fluent humour – "polished and agreeable" in the OED's definition. In this context it would naturally refer to comedy, and The Two Gentlemen may well have been in Chettle's mind when he wrote this in the latter months of 1592.

Those turbulent histories and tragedies show Shakespeare writing under the shadow of Marlowe and Peele, but the literary antecedents of The Two Gentlemen are very different. One influence is the dapper court comedies of John Lyly, such as Midas and Gallathea, performed by boy-actors in the late 1580s. Another is the fashion for sentimental prose romances, pastorales and novellas – Lyly's Euphues and Sir Philip Sidney's Arcadia are the best known of these, but even more in vogue were continental imports such as Jorge de Montemayor's romance Diana, originally published in Spanish in the 1550s. There are strong traces of this work in The Two Gentlemen. Specific echoes suggest Shakespeare used the English translation by Bartholomew Yonge: if so, he seems to have read it in manuscript – Yonge's translation did not appear in book form until 1598, but as we learn from its preface, it had "lyen by [him] finished" for 16 years prior to publication.

The play has its flaws, but in a sense they are part of its fascination. While the technical mastery of his mature comedies leaves one gasping, The Two Gentlemen has a kind of guileless transparency; it lets us in on its secrets. We see the young (or youngish) Shakespeare at work, the apprentice poet carefully blending the ingredients of love, confusion and drollery that will become familiar in subsequent comedies. Julia disguising herself as a boy in order to pursue her errant lover, Proteus, is the first of many cross-dressing Shakespeare heroines – Portia, Rosalind, Viola, Imogen – though I think she is the only one to discuss the pros and cons of wearing a codpiece as part of her costume.

The play's transition from the shallow comforts of the court, where things start to unravel, to the hardships of the forest, where they are paradoxically resolved, foreshadows the restorative virtues of the forest in A Midsummer Night's Dream and As You Like It. There are certain phrasings that sound familiar, though they are of course anticipations rather than echoes, and there are some jokes that will be thriftily recycled, such as Lance's garbling of melancholy as "allicholy", which is later repeated by Mistress Quickly in The Merry Wives of Windsor.

As befits this romantic comedy there are plenty of love letters, love songs ("Who is Silvia, what is she …?", later set by Schubert) and love poems flying around. They are somewhat pastiched for comic effect, but it's notable that one of the subject-matters which fires the young Shakespeare's poetry is poetry itself, as in this lovely Ovidian riff on the subject of Orpheus' lute, "Whose golden touch could soften steel and stones, / Make tigers tame and huge leviathans / Forsake unsounded deeps to dance on sands." And, in another register, we find the instinctive deployment of erotic double entendres is already marked. Lance's "cate-log" of his sweetheart's virtues – her skills at "milking", "fetching", "spinning" etc – is a virtuoso profusion of smutty puns.

Proteus's clownish servant Lance is perhaps the best-known character in the play, though the chief reason for this is undoubtedly his mute comic stooge, Crab, whose name does not feature in the Dramatis Personae because he not a person but a dog. Lance's doting on this allegedly ungrateful hound provides a comic subtext on the main-plot themes of friendship and fidelity (this echoing of themes is itself another characteristic Shakespearean device here tried out for the first time). Crab has been the subject of Lance's devotion since he was a puppy: he was "one that I saved from drowning, when three or four of his blind brothers and sisters went to it … Nay, I'll be sworn, I have sat in the stocks for puddings he hath stolen, otherwise he had been executed; I have stood on the pillory for geese he hath killed, otherwise he had suffered for't." And now Crab has disgraced himself (off-stage) by urinating on the beautiful Silvia, and Lance is in trouble again. "Did not I bid thee still mark me and do as I do? When didst thou see me heave up my leg and make water against a gentlewoman's farthingale?" It is sometimes suggested that Lance was written for the famous comedian Will Kempe, though the dates do not quite fit (unless the part was a later addition). In John Madden's Shakespeare in Love, we see Queen Elizabeth cheerfully giggling at the Lance-Crab double act, and the theatre owner Henslowe trying to persuade Shakespeare to include a dog in his next play, currently titled "Romeo and Ethel the Pirate's Daughter". This has no claim to scholarly weight but may well communicate a genuine point that the play – pace the critics – was a popular one.

I think the great 18th-century Shakespeare scholar Edmond Malone had it right (as he so often does) when he said that both the poetry and the comedy of The Two Gentlemen "are as perfectly Shakespearean (I do not say as finished or as beautiful) as any of his other pieces". Hazlitt recognised this as well – "there is throughout a careless grace and felicity which marks it for his". It has, in short, that certain something. If an analogy between 1590s comedy and 1960s pop music is permissible, one might say that The Two Gentlemen is Shakespeare's Please Please Me – full of freshness and sweetness, rather poignantly innocent, and already containing the promise of the great creative harvest to come.

• The Two Gentlemen of Verona is at the Royal Shakespeare theatre, Stratford-upon-Avon, from 12 July until 4 September.