First, I love this play. Not having taught it, maybe you don't feel the same, but I'm glad to help.
The play starts with the guards seeing Old Hamlet's ghost on the battlements. Elizabethans strongly believed in the supernatural. This scene sets the dark and tragic mood that will permeate the entire play.
Hamlet has returned from school for his father's funeral. (It is said he died of a snake bite while napping in the garden...which he did everyday, at the same time--perfect set-up for a murderer, huh?)
Claudius, the new king says he has decided to marry Gertrude, Hamlet's mom, though the body is "barely cold." He also announces that young Fortinbras, Denmark's enemy (in Norway), is up to "no good," and sends messengers with a communication for Old Norway, the king there, and Fortinbras's uncle. (Fortinbras and Hamlet have a lot in common.)
At the end of the scene, the guards come to tell Hamlet about the Ghost's appearance. ("Ghost" is capitalized, as that is what the character is named: if I didn't make it clear, Old Hamlet is the Ghost.)
Ophelia is Hamlet's sweetheart. As her brother Laertes leaves for school, he warns Ophelia to "remain chaste" (not to fool around with Hamlet), and she reminds HIM to also be chaste. Polonius, their father, is a pretty foolish man, but he delivers great (and famous) advice in this scene.
Hamlet goes the battlements to confront the Ghost.
Hamlet finds out from his father's ghost that his dad was murdered.
Old Hamlet's ghost has charged Hamlet to avenge the dead king's death. (Ultimately, Hamlet has to decide if the ghost is a good, real ghost, or a demon disguised to trick Hamlet to his eternal damnation: the murderer is his Uncle Claudius, the old king's brother; it is a sin to kill a king.)
The Ghost has sworn Hamlet to say nothing of what he had told his son, and tells him also not to punish Gertrude—(she, in Elizabethan eyes, is guilty of incest by marrying her brother-in-law), but to leave it to Heaven.
So Hamlet decides he will have to pretend to be mad to see if he can find out what really happened, and swears those on the battlements with him to keep his secret. (It is his continuous failure to act that is considered Hamlet's tragic flaw in the play.) Hamlet "laments" that he has been chosen to avenge his dad's death.
Some of Shakespeare's most quoted lines and speeches are in this play. Good luck. (Use the enotes summaries noted at web address listed at the bottom of this summary: there are scene-by-scene summaries, and an analysis of each scene, as well.) :)
Please see the link below for more answers.