In order to understand the rest of the play, you must understand the relationships the characters have with one another, and you must understand the dynamics within those relationships. In Scene 2 we first meet King Claudius who has taken over the throne of his recently deceased brother. He has also just recently married his brother's wife/widow, Gertrude. Prince Hamlet is the son of Gertrude and the former king, so that makes Claudius his uncle and how his step father. Hamlet is not happy about this turn of events and his first line reflects this attitude. Claudius call Hamlet his son, and Hamlet, in an aside, says, "a little more than kin and less than kind." He feels a little too closely related to Claudius now. Hamlet loves his mother but is diappointed in her decision to marry Claudius. He agrees to stay at Elisinore for her sake. This shows that Hamlet is still a dutiful son at heart. Claudius, in an attempt to show fatherly concern, gives a very long speech to Hamlet about how he needs to more quickly accept his father's death as an event that is natural and expected and therefore should be grieved and moved past. He comes across as rather cold and practical, rather than loving and understanding.
In this scene we also briefly meet characters who become more important later in the play. Laertes is the son of Polonius, and Polonius is a chief courtier and adviser to Claudius. Laertes goes through the public obligation of asking the King's permission to return to college now the the funeral and wedding are over. Claudius, wanting to maintain the support of a courtier like Polonius, is very flattering in his behavior towards Laertes , and grants the request.
This scene also shows us the friendship between Hamlet and Horatio. In scene 1, Horatio is portrayed as the learned man who verifies the appearance of the ghost. In this scene, we see the friendship between the two men and how supportive Horatio is light of all of the change in Hamlet's life. Horatio, as kindly as possible, and with a lot of patience of Hamlet's questions, reveals all of the details he can about the appearance of the ghost of King Hamlet. He is rightly concerned about Hamlet's wanting to see it for himself, but he is supportive nonetheless.