In Romeo and Juliet -Act 2, scene 2, why does Romeo not tell Juliet he's there at the very beginning? What is he doing instead?

Act II, Scene 2 of Romeo and Juliet is the famous balcony scene where the two offspring of the Montagues and Capulets express their love for each other. In Act I we are introduced to both characters and understand the stresses in their lives. They come from families which are embroiled in a bitter feud, Romeo is in love with a woman who does not reciprocate his love, and Juliet is being encouraged to marry a man she barely knows.

The two meet when Romeo and his friends "crash" the Capulet's party. Romeo comes to the party ostensibly to see the woman he loves, "the fair Rosaline," but, once he sees Juliet from across the room he immediately forgets about his previous infatuation. Shakespeare wants you to believe that it is love at first sight. In Act I, Scene 5, Romeo says,

O, she doth teach the torches to burn bright!
It seems she hangs upon the cheek of night
As a rich jewel in an Ethiop’s ear—
Beauty too rich for use, for Earth too dear.
So shows a snowy dove trooping with crows
As yonder lady o’er her fellows shows.

Likewise, once she lays eyes upon the young Montague, Juliet is smitten. At first meeting she allows Romeo to steal a kiss from her. The end of the party disrupts their meeting and at the end Act I they learn the truth of their identities. Not satisfied, Romeo ditches Mercutio and Benvolio and goes over the wall into the Capulet's orchard to see if he can catch a glimpse of Juliet. He risks life and limb in doing so because he has already drawn the ire of Juliet's cousin, the belligerent Tybalt.

When he sees Juliet up in the balcony Romeo continues the theme of comparing her to something bright:

But soft, what light through yonder window breaks?
It is the East, and Juliet is the sun.
Arise, fair sun, and kill the envious moon,
Who is already sick and pale with grief

For Romeo, Juliet is the light. Before meeting her he has been depressed and it is said that he often locked himself in his room and shut out the light. He stands below her balcony without revealing himself for several minutes admiring her and he again he notices her amazing beauty:

The brightness of her cheek would shame those stars
As daylight doth a lamp; her eye in heaven
Would through the airy region stream so bright
That birds would sing and think it were not night.

It is not clear whether Romeo would have ever spoken to Juliet if he had not heard her refer to him as she muses quietly to herself. Romeo listens without speaking as Juliet utters the famous lines and laments the fact that Romeo is a Montague:

O Romeo, Romeo, wherefore art thou Romeo?
Deny thy father and refuse thy name,
Or, if thou wilt not, be but sworn my love,
And I’ll no longer be a Capulet.

Finally, Romeo speaks up and the plot is propelled forward as they make arrangements for marriage in some of the most beautiful language ever written by Shakespeare. The scene has been performed and filmed many times. The best of these is probably Zeffirelli's 1968 version which really captures the youthful enthusiasm of the two characters. It also gives a good picture of what the Capulet estate might look like and how Romeo enters and stands under a tree while spying Juliet on the balcony before the two speak.