How are hate and revenge displayed in Romeo and Juliet?What actions and quotes are said to prove that there is hate and revenge in the play?

There are many quotes and points of action in the play
that reveal the hatred between the Capulets and Montagues and cries for and actions of
revenge.  So, I will focus on a main character whose sole function could be said to be
to display the hatred of the Capulets for the Montagues --
Tybalt.

In Act I, Scene i, Tybalt happens upon the servants
of both houses brawling in the streets, and rather than "keep the peace" as ordered by
The Prince, he pours oil on the fire by taunting
Benvolio:

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What drawn and talk of peace?  I hate the
word,

As I hate hell, all Montagues, and
thee.

Later in the Act at the
Capulet party, he recognizes Romeo and tells Lord Capulet that he will call Romeo out
then and there.  Capulet wants to keep the party atmosphere and has his patience so
tried by Tybalt's insistence upon throwing Romeo out that he loses his temper and
commands Tybalt to ignore Romeo.  Tybalt exits the scene with a nice foreshadowing of
the duel to come with Romeo:

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I will withdraw.  But this intrusion
shall

Now seeming sweet, convert to bitt'rest
gall.

And finally, he is so
bent on revenge against Romeo for besmirching his family's gathering, that he takes up
Mercutio's challenge when Romeo refuses to fight.  Once Tybalt slays Mercutio, he's
still ready for Romeo, inflaming Romeo's anger by calling him "boy."  They fight, and
Romeo kills Tybalt and must flee.

In Romeo and
Juliet, Tybalt is the closest thing to a villain in the play.  This is
important because, in order for Romeo to remain the hero the audience is rooting for,
Tybalt should be seen as an instigator of the "evil" actions of the play.  He should
appear to the audience as someone who bears the responsibility for the duel that goes
against the ruling for peace and causes Romeo's banishment.