If you're interested in a look at what Elizabethan theatrical "stardom" looked like, check out Richard Burbage. Burbage originated the role of Hamlet, which seems to have rocketed him into the public eye -- he came from a theatrical family and made his fortune not solely through acting but also through a variety of business interests related to the theater, so in modern terms he might be comparable to an actor turned director/actor turned producer who just kept acting. Richard Burbage is associated with many of Shakespeare's most prominent male protagonists -- Othello, Lear, Henry V, Macbeth, Coriolanus -- and his performances were praised by Shakespeare's contemporary playwrights, but in terms of popularity among the general public he seems to have been a major audience draw. The general public went ape when Burbage died in 1619, as did playwrights and poets associated with the theater and who perhaps knew him personally as well as knowing his performances -- all the effusive elegies and memorials to Burbage are a little reminiscent of the hagiographic responses to deaths of modern actors, and inspired one anonymous author to scorn:

Burbage, the player, has vouchsafed to die!

Therefore, in London is not one eye dry:

The deaths of men who act our Queens and Kings,

Are now more mourn'd than are the real things.

In general, though, audience members who weren't otherwise involved in the arts made their opinions on actors known when they paid money to see a certain performance. Word-of-mouth was a factor in whether a play succeeded or failed, no doubt as much word of mouth concerning the actors and performances as well as exciting stage tricks or the bare content of the play, and it's easier to document what Ben Jonson or another playwright thought about such-and-such leading actor than it is to find documentation of what Q. Random Groundling thought of that same actor's performance.

Another famous actor and contemporary of Shakespeare might be Edward (Ned) Alleyn -- Alleyn was associated with the male leads of Christopher Marlowe's works in the same way Burbage was associated with Shakespeare's, and was associated with his performances as the title characters in The Jew of Malta, Doctor Faustus, Tamburlaine, etc. Alleyn too made a pretty penny both from his work as an actor and his shrewd investments in the theater -- his performances were memorable enough that later in life he reprised the roles he'd played in Marlowe's plays to drum up audience interest in a new theater he had a stake in. He seems to have been a competitor of Richard Burbage; Bart van Es' post on Burbage as Hamlet for the OUP blog goes into the rivalry (friendly or not-so-friendly) between Burbage and Alleyn a little.

Will Kempe was an actor associated with Shakespeare's company who specialized in comedic roles -- he may have originated the part of Falstaff and is associated with Peter in R&J and Dogberry in Much Ado About Nothing. He was especially famous for his skill at dancing (itself a strong component in Elizabethan drama -- a couple modern companies preserve this, if you've seen Globe performances on video where they end in a big dance number that's an echo of that) and people came to plays hoping to see him dance. At the end of his life, Kempe was succeeded in Shakespeare's talent pool by another comedic actor, Robert Armin. (In general, the character of Falstaff seems to have been a massive crowd-pleaser, and there's a theory that Falstaff had at least one high-profile superfan -- Queen Elizabeth I, whose request for a play dealing with Falstaff in love was the ostensible impetus for his appearance in Merry Wives of Windsor.)

There are other famous actors we don't necessarily know much about, like Samuel Cross -- generally we know they're famous because other actors describe them as such, as the big names and box-office draws they might seek to emulate or supplant, or because playwrights mention them as actors they might have written for. Shakespeare also acted in his own plays, but generally not in leading roles. Modern accounts of who played what in specific Shakespeare plays, barring big names like Burbage who were indelibly associated with their best-known roles, are mostly speculative; one aspect of Shakespeare scholarship that's a massive rabbit-hole for historians and theater fans is trying to identify which roles were tailored to which performers. Acting companies in Shakespeare's social circle seem to have been an all-hands-on-deck affair -- actors doubled roles, playwrights sometimes doubled as actors, senior actors held shares in the company and profited from its success. These shareholders might hold financial stakes in the construction and operation of new playhouses, and they might have a hand in the recruiting of new talent; in general, if you wanted to make a lot of money and to keep acting, you'd want to be positioned more like Burbage. But child actors and actors specializing in women's parts don't seem to have had a lot of financial leverage at least during that span of their career, perhaps due to the comparably short span in which they'd be able to reprise those roles; it's harder to say who the equivalent of a Shakespearean leading lady might be. Some senior actors may have graduated to playing leading men after first having careers as boy-actors, while other boy-actors may have washed out.