Cyrano doesn't even appear until about half-way through Act I, but when he does, not only does he appear from the audience (when the play is performed), but he's yelling at one of the actors to get off the stage because, as we find out shortly afterward, said actor had been hitting on Cyrano's girl. Actually...all of Act I is Cyrano's Establishing Character Moment.
The opening scene, when we find out that Antonio is chronically depressed, Salarino and Solanio are Those Two Guys who want to cheer him up, Gratiano is a party animal who constantly goes off on tangents and Bassanio is Antonio's Heterosexual Life-Partner who's blown a lot of money and wants to marry the fair Portia. (Oddly enough, Lorenzo, the other guy in the scene, doesn't really get his character established until later—although he does make a crack about Gratiano never letting him speak, and later on he chastises Launcelot for talking too much, so presumably talkative people get on his nerves.)
Portia and Nerissa's first scene sets Portia's personality up for the rest of the play, besides establishing Nerissa as her best friend and straight woman.
Shylock's status as a Jew and a usurer, plus his grudge against Antonio, are set up in his first scene, when he and Antonio argue about the theological implications of usury. Plus, he makes the "pound of flesh" bargain at the end of the scene. The Character Development that turns him into an Ineffectual Sympathetic Villain—and leads to his famous "Hath not a Jew eyes?" speech—comes in dribs and drabs.
The Prince of Morocco gets his character established as a precursor to his choice of the wrong casket. Arragon's speech also serves this purpose later on, but he's only in one scene...
Launcelot. "Certainly my conscience will serve me to run from this Jew, my master!"
Jessica gives a soliloquy letting everyone know that she's Shylock's daughter, that she's ashamed to be his daughter, that she's ashamed of herself for being ashamed, that she's nothing like him, and that she's in love with Lorenzo and wants to marry him and become a Christian.
The two scenes in which Lorenzo a) sets up his elopement with Jessica and b) carries it out.
The whole of Denmark is celebrating the new king's wedding yet Hamlet is sulking in a corner wearing black because he's still mourning his father.
The opening scene of Company does this for several of the characters. First, it shows Bobby's popularity and his friendship with the couples. Sarah and Harry's competitive and combative relationship is established as they give him their present. Amy's neurotic nature is shown as she presents him with the present from her and Paul. Joanne's cynical and sarcastic nature is shown when she snarks at the other wives and, when one of them asks Bobby who she is, responds thus:
Joanne: That is I, miss. I'm very rich and I'm married to him, and I'd introduce him but I forgot his name.
The musical version of Heathers quickly establishes all the main characters (except J.D., who turns up a couple scenes later) in its opening number "Beautiful." It establishes Kurt and Ram as Jerk Jocks, Martha as a Butt-Monkey (but a lovable one who is a "sucker for a happy ending"), Heather Chandler as a "mythic bitch", Heather Duke as the Butt-Monkey of her own Girl Posse, Heather McNamara as the nicest of the trio, and Veronica as the wry, observant outsider who just wants to fit in.
J.D. gets his, too: sitting quietly on the outskirts with a book, making a snarky, semi-flirty comment to Veronica, but otherwise minding his own business, until Kurt and Ram start harassing him.
Ram: Hey, Kurt, didn't this cafeteria have a "no fags allowed" rule? J.D.: Yeah, but they seem to have an open-door policy on assholes, don't they? (beat) Kurt: ...Hold his arms. (Ram attempts to do just that, but J.D. beats the crap out of them both single-handed)
Hamilton has an excellent one for Aaron Burr that perfectly encapsulates his cautious, noncommittal personality from his very first line.
Hamilton: Pardon me. Are you Aaron Burr, sir?
Burr:That depends. Who's asking?
The Takarazuka version of Roméo et Juliette, de la Haine à l'Amour establishes the quarreling characters of Tybalt, Mercutio, and Benvolio all within 3 seconds, with Tybalt spitting on Benvolio as he holds his hand out for a shake, Mercutio rushing infuriated to his friend's defense, and Benvolio holding Mercutio back. Meanwhile, Romeo is introduced with a dazed smile and smelling a flower.