When Benvolio, a Montague and Romeo's cousin, arrives on the scene, he tries to calm down the furor but is then interrupted by Tybalt who threatens him:

What, drawn and talk of peace? I hate the word
As I hate hell, all Montagues, and thee.
Have at thee, coward!

When the brawl is broken up by the Prince's men the Prince comments on the bitter feud and the violence which is part of it:

Rebellious subjects, enemies to peace,
Profaners of this neighbor-stainèd steel—
Will they not hear?—What ho! You men, you beasts,
That quench the fire of your pernicious rage
With purple fountains issuing from your veins  

An excellent summation of the feud comes from Romeo as he enters the scene after the violence. He says that each side continues the feud simply because they love fighting:

O me! What fray was here?
Yet tell me not, for I have heard it all.
Here’s much to do with hate, but more with love.

Later in Act I when Tybalt discovers Romeo at Capulet's party he is angered and wants to challenge the Montague. The fuming Tybalt is dissuaded by Capulet from fighting, but his indignation is apparent and he says he will do something about it later:

Patience perforce with willful choler meeting
Makes my flesh tremble in their different greeting.
I will withdraw, but this intrusion shall,
Now seeming sweet, convert to bitt’rest gall.

At the end of Act I Juliet also feels the sting of the rivalry as she finds out that Romeo is a Montague and laments falling in love with an enemy of her family. She says,

My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me
That I must love a loathèd enemy.

Benvolio's words at the beginning of Act III again demonstrate the bitterness of the feud as he warns Mercutio to get off the street before the Capulets come along:

I pray thee, good Mercutio, let’s retire.
The day is hot, the Capels are abroad,
And if we meet we shall not ’scape a brawl,
For now, these hot days, is the mad blood stirring.

When Tybalt comes on the scene he challenges Romeo. In a case of dramatic irony, because Romeo has already married Tybalt's cousin, Romeo backs down. Tybalt is not to be put off and he says,

Romeo, the love I bear thee can afford
No better term than this: thou art a villain.

There are numerous more quotes that demonstrate the ferocity of the rivalry throughout other Acts of the play as well.