In Hamlet, what are the most important things to consider or help us to understand both Laertes and Ophelia, for example in Act 1 Scene 3 ?
The King has already given his permission, and so Laertes is about to go back to France. Just before he leaves, he has a little, loving and concerned conversation with his younger sister, Ophelia. Most of what he says to her is a warning about Hamlet. Laertes knows all too well that his sister and Hamlet have been lovers, and he doesn't like the relationship. He believes that it is dangerous for Ophelia to be involved with Hamlet, and will be even more dangerous in the future.
What he says to her, in essence, is this: Hamlet is not free to do as he chooses; he may say he loves you now, but because of who he is, the position he holds in the royalty of the Danish court, responsibilities may well call him away from you and you will be cast aside as one would do a trifle. In short, Laertes tells Ophelia to cut off the relationship before it is too late for her. She agrees to do so and encourages her brother, when he goes back to France, to heed his own advice. He tells her he will.
What we see here is a loving brother and sister who care for each other's welfare.
This scene introduces the King's minister, Polonius; along with his son and daughter. The pecking order becomes apparent. Laertes tells Ophelia what to do, Polonius tells Laertes what to do, and then for good measure, tells Ophelia what to do. You learn that Laertes is off to Paris to attend a university; and that Ophelia is being courted by Prince Hamlet, a circumstance that neither her brother or father see as favorable to her. Laertes tells her somewhat patronizingly, not to take Hamlet seriously because he is not free as a prince to marry whom he chooses, and to safeguard her heart and her chastity. Ophelia replies archly to Laertes: "I will the effect of this good lesson keep as watchman to my heart: but good my brother, do not as some ungracious pastors do, show me the steep and thorny way to heaven, while (you) ---the primrose path of dalliance treads, and recks not his own rede". In other words, I will be a good girl if you are careful to take your own advice. It is the only instance in the play where Ophelia displays some spunk or wit.
After seeing Laertes aboard his ship, with an extended directive on how to behave in Paris, Polonius turns his attention to Ophelia, and in harsher terms, repeats the directive against Hamlet. The scene ends with Ophelia's consent to avoid Hamlet.
Laertes and Ophelia are on good terms as siblings, both somewhat henpecked by their overbearing father. The scene is key as it not only introduces them, but advances the action and defines the character of Polonius.