The first instance of revenge that affects the characters in Shakespeare's Romeo and Juliet occurs long before the play begins.

CHORUS. Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean. (Prologue, 1–4)

The reason for the "ancient grudge" and the first act of revenge between the Capulets (Juliet's family) and the Montagues (Romeo's family) is never explained in the play, if anybody even remembers what the reason is.

Nevertheless, the feud seems to be in the forefront of everyone's mind. The first two characters who enter the play carry swords and bucklers (small, round shields up to 18 inches in diameter), and they're ready for a fight. Within a few minutes, there's a full-scale brawl in the streets between servants and family members of the Capulets and the Montagues.

The Prince appears on the scene to break up the brawl.

PRINCE. ...What, ho! you men, you beasts,
That quench the fire of your pernicious rage
With purple fountains issuing from your veins! (1.1.79–81)

The "pernicious rage" is a lust for revenge that seems to permeate the sensibility of every member, servant, and friend of each family.

Despite the Prince's warning against future brawls in the street under pain of death, it's not long before Mercutio and Tybalt are sword fighting in the street. Tybalt kills Mercutio when Romeo tries to come between them, and Romeo kills Tybalt in retaliation for Tybalt killing Mercutio.

The Prince is once again on the scene, and Benvolio explains what happened.

BENVOLIO. ...An envious thrust from Tybalt hit the life
Of stout Mercutio, and then Tybalt fled;
But by-and-by comes back to Romeo,
Who had but newly entertain'd revenge,
And to't they go like lightning; for, ere I
Could draw to part them, was stout Tybalt slain;
And, as he fell, did Romeo turn and fly. (3.1.173–179)

Lady Capulet calls for revenge against Romeo.

LADY CAPULET. I beg for justice, which thou, Prince, must give.
Romeo slew Tybalt; Romeo must not live. (3.1.185–186)

The Prince decides against ordering Romeo's death for his part in the sword fight and for killing Tybalt, but banishes Romeo from Verona.

Romeo spends his wedding night with Juliet before leaving Verona for Mantua. Lady Capulet comes to Juliet after Romeo hurriedly exits through a window in Juliet's room. Lady Capulet is not satisfied that the Prince has exiled Romeo from Verona and vows revenge for Tybalt's death.

LADY CAPULET. We will have vengeance for it, fear thou not.
Then weep no more. I'll send to one in Mantua,
Where that same banish'd runagate doth live,
Shall give him such an unaccustom'd dram
That he shall soon keep Tybalt company;
And then I hope thou wilt be satisfied. (3.5.90–95)

Paris also gets caught up in the feud and confronts Romeo at Juliet's tomb. Paris believes that Romeo has come to desecrate the tomb, and Paris also seeks his own revenge against Romeo for what he thinks is Romeo's part in Juliet's death, which prevented Juliet from marrying Paris.

PARIS. This is that banish'd haughty Montague
That murdered my love's cousin—with which grief
It is supposed the fair creature died—
And here is come to do some villanous shame
To the dead bodies. I will apprehend him.
Stop thy unhallowed toil, vile Montague!
Can vengeance be pursu'd further than death?
Condemned villain, I do apprehend thee.
Obey, and go with me; for thou must die. (5.3.49–57)

The deaths of Romeo and Juliet finally bring an end to the cycle of revenge.

In act 1, scene 1, Tybalt, always a firebrand, sneers at Benvolio, who is trying to keep the peace, saying,

What drawn and talk of peace? I hate the word, / As I hate hell, all Montagues, and thee.

Tybalt seems to have no idea why he has such feelings of hate and vengeance towards the Montagues, but he is willing to street brawl with them over a feud whose cause seems to have been long forgotten.

In act 1, scene 5, Tybalt sees Romeo flirting with his cousin Juliet at the masquerade ball. After Lord Capulet advises him not to start trouble at the party, Tybalt vows revenge:

I shall withdraw but this intrusion, now seeming sweet, shall convert to bitt'rest gall.

In fact, Tybalt, though he loses his life in the process, does manage to convert Romeo and Juliet's relationship into bitter gall or tragedy—however, the tragedy does end the feud at last.

In act 3, Prince Escalus, dismayed at the renewed violence in which Tybalt has killed Mercutio and Romeo has killed Tybalt, states:

I have an interest in your hate's proceeding, / My blood for your rude brawls doth lie a-bleeding; But I'll amerce you with so strong a fine / That you shall all repent the loss of mine.

The prince is a voice of reason, trying to end the cycle of revenge and death.

In act 5, at the end of the play, as the families and Prince Escalus mourn the deaths of Romeo and Juliet, the prince again speaks. He tells them that losing so many of their loved ones is the fruit of their vengeful hate and blames himself for not acting sooner to end the fighting:

Capulet! Montague!

See what a scourge is laid upon your hate,

That heaven finds means to kill your joys with love!

And I, for winking at your discords, too

Have lost a brace of kinsmen.

Here are 5 instances of a character seeking revenge in "Romeo and Juliet":

1. At the party for Juliet, Tybalt learns that Montagues have slipped in:

This, by his voice, should be a Montague/Fetch me my rapier, boy.  What dares the slave/Come hither, Cover'd with an antic face,/To fleer and scorn at our solemnity?/Now by the stock and honour of my kin,/To strike him dead, I hold it not a sin (I,v,51-56)

2. Mercutio seeks to avenge the Montagues against the insults of Tybalt:

Oh, clam, dishonorable, vile submission!/Alla stoccata carries it away....Good King of Cats, nothing but one of your nine lives, that I mean to make bold withal....(III,i, 61-66)

3. After Mercutio is slain, Romeo vows revenge against "Tybalt's slander":

O sweet Juliet,/Thy beauty hath made me effeminate,/And in my temper softened valor's steel!...This day's black fatae on more days doth depend,/This but begins the woe others must end. (III,i, 96-103)

4. In Act 3, scene 5, as Juliet bemoans the banishment of Romeo, Lady Capulet enters and speaks to her daughter.  She tells Juliet their family will be avenged,

Well, girl, thou weep'st not so much for his death/As that the villain lives which slaughtered him....the traitor murderer lives....We will have vengeance for it fear thou not. (III,v, 79-88)

5. When Paris encounters Romeo in the catacombs of the Capulets, he interprets Romeo's actions as puposefully vengeful the Capulet grave and seeks redress against this crime:

Stop thy unhallow'd toil, vile Montague!/Can vengeance be pursued further than death?/Condemned villain, I do apprehend thee:/Obey and go with me; for thou must die. (V, iii, 54-57)