What quotes reveal Juliet's age, Romeo's age, and the two characters wanting to get married?

Unfortunately, there is no textual evidence to declare exactly how old Romeo is in Shakespeare's Romeo and Juliet. The Friar and the Nurse both call him a young man at different times, which suggests that he is closer to Juliet's age than he is to making a name for himself out in the world. Another clue about Romeo's age might be as Lord Montague confides with his nephew, Benvolio, about his concerns for Romeo as follows:

"The shady curtains from Aurora's bed,

Away from light steals home my heavy son,

And private in his chamber pens himself,

Shuts up his windows, locks fair daylight out,

And makes himself an artificial night" (I.i.129-133).

The above passage proves that Romeo still lives at home with his parents just as a teenager would. His father notices that his son blocks daylight out of his life, or else he is sleeping most of the day. Again, such is the life of many a teenager. A specific age for Romeo, however, is never explicitly revealed.

As far as Juliet's age is concerned, though, Lord Capulet reveals her age to Paris in the following passage:

"By saying o'er what I have said before.

My child is yet a stranger in the world;

She hath not seen the change of fourteen years.

Let two more summers wither in their pride

Ere we may think her ripe to be a bride" (I.ii.7-11).

This proves that Juliet is 13 years old, going on 14. It also demonstrates that her father does not wish her to marry so young because she is still "a stranger in the world." Little does he know that Juliet will get married very soon.

On the night of the Capulet masquerade ball, Romeo and Juliet unexpectedly meet at the party and then later outside her balcony. It is here that they fall deeper in love and make wedding plans. Juliet mentions marriage first:

"If that thy bent of love be honourable,

Thy purpose marriage, send me word tomorrow,

By one that I'll procure to come to thee,

Where and what time thou wilt perform the rite" (II.i.185-188).

Romeo tells her he will send someone by the hour of nine to inform her of how the wedding will happen. He immediately goes to Friar Laurence to secure him to perform the marriage. He says to Friar Laurence, "We met, we wooed, and made exchange of vow/ I'll tell thee as we pass; but this I pray,/ That thou consent to marry us today" (II.ii.62-64).