How can I come up with a thesis and three arguments about the immaturity of the characters in Romeo and Juliet?

Although it may be easy to assign a good deal of immaturity to the character of Juliet, because she is only 13, Romeo and Mercutio are easily the most immature characters in Shakespeare's tragedy.

While Juliet makes the tough decisions, Romeo, on the other hand is quick to act without thinking and ignores the advice of Friar Laurence who says in Act II, Scene 6,

"Therefore love moderately; long love doth so; Too swift arrives as tardy as too slow."

Romeo fails to heed the friar's words and at almost every juncture of the play acts rashly without fully considering his path. Even at the beginning of the play we meet a Romeo who is badly wounded by Cupid's arrow and throughout his opening lines in Act I he moans and groans about how he is in love with a woman who will not reciprocate his love. He shuts himself up in the dark of his room and takes long walks unaccompanied to wade in his sorrow.

Once he sees Juliet at the Capulet party he immediately forgets Rosaline and falls head over heels for the young girl he admires from across the room. Juliet, although very much attracted to Romeo, professes that maybe she has been too familiar and asks Romeo to wait. In the famous balcony scene, Act II, Scene 2, Juliet says,

Well, do not swear. Although I joy in thee,
I have no joy of this contract tonight.
It is too rash, too unadvised, too sudden,
Too like the lightning, which doth cease to be
Ere one can say “It lightens.” Sweet, good night.
This bud of love, by summer’s ripening breath,
May prove a beauteous flower when next we meet.

Romeo, however, is not to be put off. He presses Juliet with a marriage proposal, "Th’ exchange of thy love’s faithful vow for mine." He confers with Friar Laurence and sets up the wedding the following day. He has known Juliet for less than 24 hours.

After the marriage he tries to be the mature young man. He puts off the challenge of Tybalt in Act III, Scene I:

Tybalt, the reason that I have to love thee
Doth much excuse the appertaining rage
To such a greeting. Villain am I none.
Therefore farewell. I see thou knowest me not.

He even tries to stop the fight between Mercutio and Tybalt but because of his actions his best friend is mortally wounded under his arm. After it is revealed that Mercutio is dead Romeo loses control and once again shows his immaturity. Instead of taking a step back and analyzing the situation he launches into a fight because he feels that he has been too "effeminate" and that fate wills him forward.

Afterward he is at his most immature and obnoxious in Friar Laurence's cell when he laments the fact that he has been banished and will not be able to look upon the face of Juliet again. He ignores the fact that the Prince only sentences him to exile and not death:

’Tis torture and not mercy. Heaven is here
Where Juliet lives, and every cat and dog
And little mouse, every unworthy thing,
Live here in heaven and may look on her,
But Romeo may not.

Finally Friar Laurence assuages Romeo's fears and the young man goes off to consummate his marriage with Juliet and then to Mantua to fulfill his sentence of exile.

Of course, when he learns from Balthasar that Juliet is dead he again demonstrates his immaturity by insisting on taking his own life to be with his love. He invokes the specter of fate as he vows to lie with Juliet: "Is it e’en so?—Then I defy you, stars!" He rushes to the Capulet's vault without consulting anyone. A simple visit to the Friar would have revealed the plan to reunite the young lovers. Instead, Romeo forces his way into the tomb, kills Paris and then himself just as Juliet is awakening. (In an aside, I absolutely love Baz Luhrmann's  take on the play when he has Romeo recognizing that Juliet is still alive as he dies. It serves him right.)

Romeo is not alone in his immaturity. His best friend Mercutio consistently shows a lack of maturity. His Queen Mab monologue in Act I, Scene 4 reaks of a childlike mind with its focus on fantasy. Again, after Romeo ditches his friends to see Juliet, Mercutio cannot accept Romeo's disappearance and cries out in the night, insulting his friend:

Romeo! Humors! Madman! Passion! Lover!
Appear thou in the likeness of a sigh.
Speak but one rhyme and I am satisfied.

In Act III, the turning point of the play, Mercutio is at his most immature. Instead of heeding the warnings of Benvolio:

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

he let's Tybalt goad him into a fight which ultimately leads to his demise. (In another aside: if you watch the Zeffirelli movie Mercutio is totally out of control and let's his pride and masculinity get in the way of good sense. Of course the fight with Tybalt is probably the best sword fight ever put to celluloid.)

I think it is also possible to make a case that Lord Capulet is immature in his treatment of his daughter, especially at the end of Act III when he promises his daughter to Paris. His berating of Juliet when she refuses his declaration at the end of the Act does not reveal a man of patience and maturity. In analyzing Lord Capulet, however, I tend to like him. I like his kind words about Romeo in Act I at the party and I think he is only trying to look out for his child.

Tybalt could also be considered as an immature character were it not for the fact that Shakespeare portrays him as a villain and an antagonist to the protagonist Romeo. Tybalt is a static character and a slave to Shakespeare's plot. Romeo and Mercutio, however, should know better.