This is a great question and a heavily-discussed one! There's kind of a commonplace way of describing Shakespeare's audiences that constructs a divide in the audience -- the working-class groundlings down in the pit munching on oranges and hazelnuts, there to see sword fights and dick jokes and maybe throw some rotten tomatoes, and a handful of aristocrats parked on velvet cushions stroking their chins thoughtfully at the complex symbolism. The reality was probably more of a tossup, with most of the audience falling somewhere between those extremes of sophistication. Shakespeare's plays have a reputation for being highbrow elite entertainment but Shakespeare was one of many playwrights in his own day, and theater was one of several readily-accessible options for public entertainments. Theater ranked alongside bull- and bear-baiting and bowling for popular public entertainment, significant enough and crowded enough to pose a perceived danger to the public in times of plague. (On top of that were other pastimes, of course -- sporting matches, eating and drinking, shopping, going to church, hitting up a brothel -- but playhouses, animal-baiting houses, and bowling alleys were considerably easier to shutter for public-health reasons.) Playhouses were more accessible to the urban lower and middle classes than pastimes that required long-term resource commitments and lots of land in the country; they could be zoned for the suburbs or the city, they could be accessed at a variety of price points, they could be built to simultaneously accommodate a diversity of viewing styles while keeping upper-class viewers as safely sequestered from lower-class viewers as they wanted to be. These playhouses weren't completely egalitarian, but they catered to a mix of classes out of economic necessity, and they could be pretty profitable for the collaborators who ran them.

Elizabethan and Jacobean folks went to the theater a lot, and they participated in the arts as spectators across class lines. The demographics of a typical London theater's audience looked a lot like the demographics of London itself -- old people, young people, wealthy ladies and noblemen, well-off men of the town and their wives and daughters, financially stable tradespeople, street vendors, children, beggars, thieves, vendors, and sex workers, with occasional visits from foreign dignitaries. It's possible that certain playhouses tried to court certain demographics -- individual playhouses may have put on shows calculated to attract rowdy young men-about-town or have knowingly accommodated sexual commerce between audience-members, or conversely they might have raised prices and enforced strictures on attendance to attract a more highbrow crowd -- but going to the theater didn't have the class connotations of, say, a night at the opera or catching a Broadway show today. Women attended the theater and were perceived as engaging with drama in a different way than men due to their different natures; to address the audience as "gentlemen" and "gentlewomen" or even just "gentles" was to invoke both a stylized kind of class courtesy and to invoke the tender sentiments of Shakespeare's spectators, especially the ladies. The playhouse itself could be a somewhat unruly space, but not so unruly that theater itself became a no-go entertainment; noblemen and noblewomen up to and including the Queen herself could commission private performances (sometimes with messy results, as with the private performance of Richard II commissioned to accompany the Essex rebellion of 1601) and at least one of Shakespeare's plays (Henry VIII) premiered as a special event for the wedding guests of King James I's daughter, complete with bombastic special effects. The possibly-apocryphal tale of Queen Elizabeth I being a high-profile fan of Falstaff and requesting a play about Falstaff in love (as the impetus for the creation of Merry Wives of Windsor) gets its zestiness from both the strength of Queen Elizabeth I's personality as a much-courted patron of the arts and the idea of Elizabeth as just another audience member susceptible to Falstaff's charms.

Elizabethan and Jacobean peoplewent to the theater a lot, they went to see the works of many playwrights in many genres, and they weighed in with their money -- like modern filmgoers, it's a reasonable surmise that people paid to see new and flashy entertainments as well as favorite performers and that if a show was especially pleasing they might pay to see it twice. Shakespeare and his contemporaries would have been well aware that their works necessarily needed to please their audiences, rather than solely pleasing patrons, impresarios, or fellow writers. In the prologue of Ben Jonson's Epicoene, Jonson remarks on a certain sect of writers who snobbishly shun everything popular in theater and seek to please only themselves with their works, only to say that he's not that kind of writer:

Our wishes, like to those make publick Feasts,

Are not to please the Cooks taste, but the Guests.

Jonson goes on to elaborate on the metaphor of a public feast by saying this feast will serve a variety of dishes suited to a variety of palates, with nothing critical missing from the table -- there will be delicacies fit for lords and ladies, serving-wenches and stylish women of the town, and he even generously allows there'll be material for the "men and daughters of Whitefriars", a notorious red-light district. If you wanted to please a narrower pool of individuals and remain financially solvent while doing so, you'd be better off publishing your work in another format -- theater was essentially oriented around performance, and performance was oriented around pleasing the audience.

The impulse to divide Shakespeare's plays (and his audience) into "high" and "low" goes way back in Shakespeare scholarship -- in one corner, musical numbers, bears, dumbshows, and rude humor, in the other subtle political commentary and sophisticated use of symbolism -- but it's not necessarily reflective of how people experienced Shakespeare's plays at the time they were first being performed. As a theory it seems to account for the somewhat incongruous variety of elements in Shakespeare's plays, but it doesn't account for the possibility that seemingly-superficial elements like dance sequences or dumbshows served a deeper dramatic purpose than we ascribe to them as modern viewers, or the moments in Shakespeare's plays that combine political commentary and rude humor quite deftly like the leek subplot in Henry V, or the simple fact that you don't need to be a rotten-fruit-lobbing groundling to take pleasure in a cathartic dance sequence or an onstage bear. Audience members presumably did like the elements that get repeated across plays, in accordance with individual tastes -- crossdressing, ethnic humor, bombastic speeches, tragic misunderstandings, graphic violence -- but we don't have the kind of data we'd need to differentiate in fine detail who liked what, when, and why. Shakespeare's audiences would have appreciated the individual subtleties of his work to varying degrees -- many of Shakespeare's plays get mileage out of stories that would already be approximately familiar to an Elizabethan of moderate education, like his plays set in Classical settings drawn from familiar Classical texts. Early Modern audiences across the board would be much more fluent in the stage conventions and poetic devices used by Shakespeare and his contemporaries, simply because they were immersed in them on a regular basis rather than being exposed to them in the limited context of either historical curiosities from 400-odd years ago or The Finest Literature Of All Time. Even a dedicated 21st century aficionado of theater who catches every single piece of Early Modern drama staged in their area any given year has less continuous access to that same Early Modern drama than actual Early Modern people had, and certain aspects of the plays that come off historical inside-baseball now (like allusions to current events, politics, and fads) might have been thuddingly obvious to any costermonger off the street or politically-active tailor in 1594 or 1603. Some of Shakespeare's material would be understood by any given viewer and some would be lost on them, but you could say the same for any theatergoer now, whether they're catching The Tragedy of Julius Caesar or The Death of Stalin.

I talked a little about the use of French in Shakespeare's plays in an earlier answer about Princess Katherine's swears in Henry V -- unfortunately we don't have a broad survey of multilinguality in 16th/17th century England, but I think we have reason to believe that enough audience members spoke sufficient French to appreciate these jokes and to warrant their inclusion. Shakespeare's French isn't perfect idiomatic French, but it's good enough to suggest Shakespeare himself was reasonably proficient. For Elizabethan English audiences French was easily the most popular second language after Latin -- helped along by the historic proximity between England and France and and sixteenth-century England's influx of French-speaking immigrants. Jokes in French would have been much more accessible to the average viewer than, say, jokes in Italian in Romeo & Juliet.