families were feuding seems to be of no consequence whatsoever to Shakespeare. The only thing that matters to him, and to the play, is that the families are feuding. The feud drives the play forward from the very first scene to the tragic end of the star-crossed lovers who get caught in the middle of it.

CHORUS: Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny . . . (1.Pro.1–3)

It's an "ancient grudge." That's the only thing we're told about the feud. How did it start? Nobody says. There's no discussion about it. Maybe nobody even remembers how the feud started.

Perhaps the feud started in much the same way that the recent brawling in the streets started:

PRINCE: Three civil brawls, bred of an airy word
By thee, old Capulet, and Montague,
Have thrice disturb'd the quiet of our streets . . . (1.1.85–87)

"Bred of an airy word" could mean a verbal slight of some kind from a member of one family to a member of the other family, or an insult—real or imagined—or something spoken purely in jest, but taken the wrong way.

Even at the end of the play, when the Montagues and Capulets reconcile—and when we might expect somebody to tell is how the feud began, now that it's ending—not one word is spoken about the reason for the feud:

CAPULET: O brother Montague, give me thy hand.
This is my daughter's jointure, for no more
Can I demand.

MONTAGUE: But I can give thee more;
For I will raise her statue in pure gold,
That whiles Verona by that name is known,
There shall no figure at such rate be set
As that of true and faithful Juliet.

CAPULET: As rich shall Romeo's by his lady's lie—
Poor sacrifices of our enmity!

PRINCE: A glooming peace this morning with it brings.
The sun for sorrow will not show his head.
Go hence, to have more talk of these sad things;
Some shall be pardon'd, and some punished;
For never was a story of more woe
Than this of Juliet and her Romeo. (5.3.307–322)

We'll never know how it all started, or why it came to this.

A simple answer to your question is that prejudice is expressed through physical fighting.  The family fued between the Montagues and Capulets is one of the most primary instances of prejudice.  Though the reasons for the hatred go unexplained, there is evidence of it in every scene.  To start, I encourage you to review any of the fight scenes which would provide physical evidence of how much the families hate each other.  The very opening scene of the play is an exchange of insults between the servants which leads to a brawl.

Other evidence of expressed prejudice is in how the characters speak of their enemies with such disdain.  When Juliet realizes Romeo's identity her reaction is "My only love sprung from my only hate..."  Romeo and Juliet's fear of revealing their love for one another to their parents and the secrecy with which they engage in a romance is further evidence.  The famous Act 2 Scene 2 balcony scene reiterates that if Romeo is even seen on the Capulet's property he will be killed.

When Romeo kills Tybalt, Juliet's internal conflict in Act 3 Scene 2 displays how strong the prejudice runs that she is torn between choosing her family or her lover.  She does not even take a moment to fully consider the circumstances.  Instead, she reacts emotionally and irrationally.

Good question!

Prejudice leads to conflict in many ways. The most basic is the ongoing clash between the Montague and Capulet families. In the chorus we're told the feud has been going on for centuries. Therefore, members of both families are predisposed to think badly of one another.

We see this even in Act I scene I, when Sampson and Gregory clash and Sampson calls Gregory "a dog of the house of Montague". We don't know what started this initial clash, but Sampson says he'll take any man or maid of Montague. That tells us flat out he's biased, and he knows it.

Act III scene I is just as open. The scene opens and Benvolio wants to clear out of the streets because the Capulets are around and "the mad blood stirring." To say before something's even happened that people are mad for a fight tells us that this is bias that's soaked deep into the bones of these people.

Greg

One could argue that from the first moment of the play in the prologue, Shakespeare starts to describe prejudice,

"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.
From forth the fatal loins of these two foes"

Because what he's stating there is that two families have such serious grudges against each other that people have shed blood over it, but yet at this point in time where the play happens, no one can remember the cause of these grudges.  Despite not even knowing why the families hate each other, people kill each other (and themselves) throughout the play because of this prejudice.

And at the end, through the Prince, Shakespeare is commenting on what a waste of energy and life this was,

"A glooming peace this morning with it brings;
The sun for sorrow will not show his head.
Go hence, to have more talk of these sad things;
Some shall be pardon'd, and some punished:
For never was a story of more woe
Than this of Juliet and her Romeo.

• Prince, scene iii"

This still happens today when people are prejudiced against a group of people (such as gay people, Black people, Latino people, people who follow Pagan religions, etc., etc.) for no real reason past that their families are and it's what they've been taught to do. 

Shakespeare shows the true nature of prejudice by not explaining its origin.  We don't know specifically what happened between the two feuding families.  We only see they are prejudiced against each other.  This is frequently the way prejudice works in the real world; people might not know why they feel a specific prejudice, but they still follow it.

Shakespeare goes on to show the depths of this prejudice in the many fights and violent acts between the two groups.  We see this type of hatred displayed between the characters Romeo and Tybalt or Benvolio and Tybalt.  We even see evidence of this prejudice between Romeo and Juliet when they first meet.  Juliet says "my only love sprung from my only hate." 

We see the effects of this prejudice clearly throughout the play.  One example is found in Tybalt's unwillingness to turn from violence.  Romeo attempts to diffuse the situation many times, but Tybalt will not forgive the perceived offense.  Eventually, Romeo and Juliet pay the ultimate price for the prejudice of their families.