What effect does Act 1, Scene 2 of Hamlet intend to have on the audience?

In Act One, scene two, of Shakespeare's Hamlet, we learn a great deal. While scene one sets a dark mood, scene two provides extensive exposition.

With paradoxes meant to convey mutual sorrow and joy, we learn that even while Old Hamlet's death is "green" (new), Claudius has married his brother's widow.

...With mirth in funeral, and with dirge in marriage,
In equal scale weighing delight and dole,
Taken to wife. (12-14)

Claudius acknowledges that Young Fortinbras is trying to take back lands his dead father lost—in a fair contest—to Old Hamlet. In this way, Shakespeare explains the threat of war that faces Denmark (hence the armed soldiers watching the battlements), and also introduces a foil for Hamlet. Later we hear even Hamlet mourn the fact that Fortinbras is so better able to do what he believes honors his dead father, while Hamlet is never able to move with equal determination to avenge his father's death.

Laertes requests permission to leave the Claudius' court, showing respect for his father's wishes and the King's.

We learn that Hamlet is disgusted by his mother's remarriage and Claudius and Gertrude's lack of mourning for Old Hamlet's death. Gertrude ask her son to put off his dark looks—death is a part of life. He agrees; Gertrude asks him why then he "seems" so sad. For Hamlet, greatly broken up over the loss of his father, he is clear that there is nothing "seeming" about his grief.

Seems, madam? Nay, it is. I know not seems.
'tis not alone my inky cloak...
Nor customary suits of solemn black,
Nor windy suspiration of forced breath,
No, nor the fruitful river in the eye... (79-84)
That can denote me truly. These indeed seem,
For they are actions that a man might play;
But I have that within which passeth show,
These but the trappings and the suits of woe. (86-89)

Hamlet says that it is not the "showing" of sorrow that matters, for an actor could easily so the same with no meaning. [This foreshadows the play-within-the-play—(III,ii)—when Hamlet has the players re-enact his father's death; even then he notes that the actors seem so capable of showing sorrow, while he is unable to act on the true sadness he feels.]

We are given a clear insight into the kind of man Claudius is: not because he offers his "love" as a father to Hamlet or names him heir to the throne, but because he (who is obviously unmoved by his brother's death—we will soon learn why) accuses Hamlet of unmanly and unholy behavior by grieving.

KING:
...'tis unmanly grief;
It shows a will most incorrect to heaven... (97-98)
...Fie! 'tis a fault to heaven,
A fault against the dead, a fault to nature, (104-105)

Gertrude wishes Hamlet not to return to school so soon. Alone, Hamlet wishes he could kill himself. Hamlet tells Horatio that the wedding occurred so soon after the funeral that they could have used the leftovers of the first, for the second.

Horatio reports that he has seen Old Hamlet's ghost—this introduces the theme of the disruption of natural order of the universe, for why else would the ghost roam the castle? Hamlet is told that his father's face is sorrowful and that he is dressed for war. Not sure if this is indeed his father's spirit, Hamlet declares that he will approach the ghost that night, and asks for silence from the men who have seen the apparition. This introduces the theme of secrecy.

The entire scene shows unnatural joy in the presence of sorrow, the need for secrecy, and possible threats from within and without the castle.

What effect does the opening scene of Hamlet have on the audience?

The effect produced by the opening scene is one of dread and foreboding. Our first inkling of unease comes as the guards, Marcellus and Barnardo, discuss what Marcellus calls "this dreaded sight twice seen of us." We learn that it is a ghost suddenly haunting the ramparts of the castle. Marcellus and Barnado have invited Horatio to see the ghost as confirmation that they are not hallucinating. Horatio at first disbelieves them, but when the ghost appears while he is there, he states:

This bodes some strange eruption to our state.

Marcellus adds to the sense of foreboding by asking Horatio why so many war preparations are going on:

Why such daily cast of brazen cannon
And foreign mart for implements of war

In Elizabethan days there was a strong belief in the supernatural, so the appearance of a ghost would surprise few viewers of "Hamlet."  In fact, the audience would welcome the spectre who is able to provide insight that a mere mortal cannot.  In addition, the audience would be curious to learn what revelations the ghost makes and how this spectre acts as an impetus, moving the characters to action.

In the first scene, for instance, Horatio and Marcellus wonder aloud about the ghost:  "Is it not like the king?"  Thus, the audience wonders what dastardly deeds have been committed to cause a ghost to return and seek retribution. Indeed, it is the ghost that reappears who acts as the catalyst to incite the melancholic and indecisive Hamlet into action against Polonius and others.