In Act 1, Scene 3 of Hamlet, what advice does Polonius give to Ophelia about Hamlet?

Polonius plays the role of jealous and possessive father in this scene. He is very suspicious of any promise Hamlet has made to his daughter, and urges Ophelia to remember that as prince, he is able to do things and say things that she is unable to do. He urges her to disregard any vows and promises that he has made to her, as he feels that any such vows and promises are made without true sincerity and only have the objective of trying to take Ophelia's virginity. Note what he says about Hamlet and how Ophelia differs from him:

For Lord Hamlet,
Believe so much in him, that he is young,
And with a larger tether may he walk
Than may be given to you.

It must be remembered at this time that an unmarried woman's worth was based on her virginity, and that if this was taken, her status would drop accordingly. Polonius therefore appears to be very strict when he rebukes Ophelia in this scene, but he can also be seen as doing what he has to in order to protect his daughter and make sure she can marry one day and live a happy life in Danish society.

What advice do Polonius and Laertes give to Ophelia in Hamlet, act 1, scene 3?

In Act I, scene iii, Laertes warns his sister to not take Prince Hamlet's profession of love seriously as marriages for princes are usually matters of state.  In other words, royal marriages are made for political reasons, not for love:

...Perhaps he loves you now,
...but you must fear,
His greatness weighted, his will is not his own,

Also, Laertes warns Ophelia against to restrain her affections lest she be embarrassed:  "Be wary then..."  Ophelia promises to heed his advice. 

Polonius enters and bids Laertes be prudent on his venture to Paris.  Then he asks his daughter about Hamlet and her.  When she says Hamlet has been affectionate, Polonius cautions her as did Laertes:  "Tender yourself more dearly...or you'll tender me a fool."  To this Ophelia replies that Hamlet has been honorable.  But Polonius is skeptical: 

"When the blood burns, how prodigal the soul...From this time
Be something scanter of your maiden presence."

Polonius warns his daughter against showing affection, also. However, his reasons are that Hamlet may simply be trying to seduce her with "mere implications of unholy suits.

To both her brother's and to her father's advice Ophelia promises to "abide."

In Act 1, Scene 3 of Hamlet, what advice does Laertes give to Ophelia?

In act 1, scene 3, Laertes is about to depart for France. Before he leaves, however, he offers his sister, Ophelia, some advice about her relationship with Hamlet. Firstly, he tells her not to take too seriously the affections that Hamlet has shown for her. He says that she should consider Hamlet's affections "a fashion and a toy in blood." In other words, Laertes doubts that Hamlet really loves his sister, and suspects that Hamlet's affections are a temporary consequence of youthful, hot-blooded passion. He says that Hamlet's affections for his sister will not last, and that they are only the "perfume and suppliance of a minute. / No more."

Laertes then tells Ophelia to remember that Hamlet, as a member of the royal family, is "subject to his birth," and that, therefore, whatever choice he might make as regards who becomes his wife, may not be his choice alone. The implication is that even if Hamlet did love her, he might very well have to disappoint her later when he has to choose a woman more suitable to take the position of queen.

Laertes continues by implying that Hamlet might really only be interested in having sex with Ophelia. He asks Ophelia to imagine what it would be like to have given her "chaste treasure" (her virginity) and thus her honor to Hamlet, only for him to subsequently tell her that he can't marry her because the country, and the royal family, want somebody else to be the queen. Laertes implores his sister to be afraid of this eventuality, and to bear it in mind should Hamlet ever tell her that he loves her. Laertes tells Ophelia that to remain afraid of this outcome is the best way for her to remain chaste, and honorable, while he is away.

What advice does Polonius give Ophelia about Hamlet?

When Polonius inquires of Ophelia what she had been discussing with her brother, Laertes, she tells him that they spoke about Hamlet. Polonius then states that he had heard that Hamlet had been spending some time with Ophelia. He then suggests:

... in way of caution, I must tell you,
You do not understand yourself so clearly
As it behoves my daughter and your honour.
What is between you? give me up the truth.

He tells Ophelia that she must be careful since she is not yet old enough to have even a clear understanding of herself as much as she should, being his daughter and for the protection of her honour (chastity). She is still immature and therefore naive. He demands that she provide him full details of her liaison with Hamlet.

Ophelia confesses that Hamlet had made a number of offers to her and proposed love. Polonius rejects the suggestion of Hamlet's affection with contempt and emphasizes how inexperienced Ophelia is, so much so, that she took Hamlet at his word. He asks her if she believed Hamlet's proposals and she replies that she does not know what to think.

Polonius then extends some fatherly advice to his daughter and says that she should see herself as an innocent baby who has taken Hamlet's offers as the truth. He says that she should value herself more highly or else she would create the impression that he is a fool for not having raised her properly. Clearly, the focus here is not entirely on Ophelia, but on himself.

When Ophelia tells him that Hamlet had actually made sincere promises of his love in an honourable manner, her father says that she must not talk nonsense. He uses the word 'fashion' to mean that it is something of short duration. He is implying that Hamlet is not sincere. His so-called passion for Ophelia is of a fleeting nature and will not last. 

When Ophelia tries to convince her father that Hamlet's utterances were truthful since he had vowed to heaven about his sincerity, Polonius rejects this by saying that it is a scheme similar to those used to trap woodcocks, birds which are deemed idiotic. His daughter would be foolish to fall for Hamlet's trap and become ensnared. He suggests that being young, Hamlet is ablaze with passion, but it is a fire that will soon burn out. Ophelia must not see these heartfelt promises as true passion, for they die even as they are made.

Polonius advises Ophelia not to make herself so easily available to Hamlet and must therefore treasure her time and see less of him. She must place a greater value on herself and not seem so easily available. He implores her not to believe Hamlet for what he says is only to fulfill his lust (unholy suits). Hamlet's vows are not sincere and can be broken as easily as they have been made. Hamlet's promises are mere tools to mislead her. He then commands her:

This is for all:
I would not, in plain terms, from this time forth,
Have you so slander any moment leisure,
As to give words or talk with the Lord Hamlet.
Look to't, I charge you: come your ways.

Polonius is quite direct and tells Ophelia that in truth, she must not henceforth waste any moment of her free time to talk to or discuss anything with Hamlet. He therefore forbids her to have such contact with him. She must obey his instruction and behave appropriately.

It is ironic that Polonius is so strict with his daughter at this point, for later in the play he will do a complete about-turn and use his daughter to glean information from Hamlet. He will actually encourage her to lead Hamlet on and essentially spy on the young prince so that he can report the details to Claudius.

What does Ophelia tell Polonius about Hamlet's behavior?

Ophelia tells her father exactly what Hamlet expected she would tell him. Hamlet is using Ophelia; he wants King Claudius to start being nervous about him, and he knows that if he acts weird to his girlfriend, Ophelia, she will tell her father, Polonius. And Hamlet knows that, since Polonius is an advisor to the King, he will straightaway tell Claudius everything Ophelia has told him.

Of course you could say, Why doesn't Hamlet just go kill Claudius? Why does he take this roundabout approach to revenge? Well, that a question for another time. Let's just say, for now, that Hamlet is a schemer and a planner and a thinker, more than he is a doer. Indeed, he would rather act at doing something than really do something.

Here, then is all of what Ophelia relates to her talkative old man about Hamlet's behavior, his wordless conversation with her, (from Act 2, Scene 1):

OPHELIA:
My lord, as I was sewing in my closet,
Lord Hamlet, with his doublet all unbraced,
No hat upon his head, his stockings fouled,
Ungartered, and down-gyved to his ankle;
Pale as his shirt, his knees knocking each other,
And with a look so piteous in purport
As if he had been loosed out of hell
To speak of horrors, he comes before me...
...He took me by the wrist and held me hard;
Then goes he to the length of all his arm,
And, with his other hand thus o'er his brow,
He falls to such perusal of my face
As he would draw it. Long stay'd he so.
At last, a little shaking of mine arm,
And thrice his head thus waving up and down,
He raised a sigh so piteous and profound
As it did seem to shatter all his bulk
And end his being. That done, he lets me go,
And with his head over his shoulder turn'd
He seem'd to find his way without his eyes;
For out o' doors he went without their help,
And to the last bended their light on me.

Oh, how very dramatic stuff! But it's just a little play acting on Hamlet's part. We will see later in this act and especially in Act 3, how fond Hamlet is of plays and players.

Yes, in a lot of ways Hamlet is a play about playing at acting without actually acting.

In Hamlet, what advice does Laertes give Ophelia about Hamlet, and why?

Before Laertes returns to school in France, he gives his sister his last words of advice. He realizes that she and Hamlet have been getting closer, so he warns her against becoming too attached to Hamlet. Laertes reminds Ophelia of Hamlet's noble standing in Denmark; because he is the prince, he is not free to marry whom he chooses. Instead, he must keep the needs of his country ahead of his own.

He himself is subject to his birth.
He may not, as unvalued persons do,
Carve for himself, for on his choice depends The safety and health of this whole state.

Laertes warns Ophelia that though Hamlet may act as though he loves her, it is not out of love but out of desire. He describes Hamlet's actions as temporary moments: just a short phase that the young prince is going through. He believes that once Hamlet gets what he wants from her, his attention will move on to someone else.

For Hamlet and the trifling of his favor,
Hold it a fashion and a toy in blood,
A violet in the youth of primy nature,
Forward, not permanent, sweet, not lasting,
The perfume and suppliance of a minute.
No more.

Therefore, Laertes thinks that it is best that Ophelia just stay away from Hamlet and ignore his flirtations. Because she is a dutiful sister, she listens to her brother and returns all of the gifts and letters Hamlet has sent her.

For a modern audience, Laertes's advice may seem strange, but Laertes worries that if his sister gives into Hamlet, her heart will be broken and her reputation ruined. His advice is intended to protect her from Hamlet.