Here we've listed some important moments in the order in which they appear in the play. We've also included some quotes to listen out for.

Act 1 Scene 2 - Setting the scene

As the court celebrates the marriage of Hamlet's mother (Gertrude) to his uncle (Claudius), Hamlet finds out that his father's ghost is haunting the castle. He is determined to confront it.
Hamlet If it assume my noble father's person/I'll speak to it though hell itself should gape / And bid me hold my peace

Act 1 Scene 5 - Hamlet meets the Ghost

The ghost of Old Hamlet reveals to his son that he was murdered by Claudius and demands that young Hamlet seeks revenge. Hamlet decides to feign madness in an attempt to find proof of his uncle's guilt. 
Ghost Thus was I sleeping, by a brother's hand/of life, of crown and queen, at once despatched

Act 2 Scene 2 - Claudius becomes suspicious

Polonius, chief adviser to the new king, tells Claudius that Hamlet's madness is due to unrequited love for Ophelia, Polonius's daughter, but Claudius is not convinced and plots with Polonius to spy on Hamlet. Hamlet's childhood friends, Rosencrantz and Guildernstern, have been summoned to the castle by the king and queen to keep close watch on him. A troupe of actors also arrives at Elsinore to entertain the court and are persuaded to include some lines written by Hamlet.
Hamlet The play's the thing/Wherein I'll catch the conscience of the king.

Act 3 Scene 1 - Hamlet's turmoil

Claudius and Polonius eavesdrop when Hamlet meets Ophelia. Hamlet speaks the famous soliloquy ('To be or not to be') and rejects Ophelia, whom he had previously professed love to, believing that she is in league with his uncle and Polonius.
Hamlet Get thee to a nunnery, go, farewell. Or if thou needs must marry, marry a fool, for wise men know well enough what monsters you make of them.

Act 3 Scene 2 - The play

In front of the king and the whole court, the players perform a scene which closely follows the Ghost's account of his murder by Claudius. When Claudius cuts the performance short, Hamlet takes this a proof of his guilt. He is now determined to revenge his father's death immediately.
Hamlet Now could I drink hot blood / And do such bitter business as the day / Would quake to look on.

Act 3 Scene 3 - Claudius's plotting

Claudius arranges Hamlet's banishment to England with Rosencrantz and Guildernstern. He confesses his crime in prayer. Hamlet finds Claudius on his knees praying but cannot bring himself to kill him there. 
Claudius O, my offence is rank, it smells to heaven: / It hath the primal eldest curse upon't / A brother's murder.

Act 3 Scene 4 - The closet scene - Hamlet kills Polonius

Unable to kill his uncle, Hamlet challenges his mother about her relationship with Claudius and admits he has been feigning madness. Hearing a noise, he believes that Claudius is eavesdropping, draws a sword and kills Polonius.
Gertrude O what a rash and bloody deed is this!

Act 4 Scene 2 - Hamlet is banished

Claudius arranges Hamlet's immediate tranportation to England and sends a letter to the English king demanding that he arranges Hamlet's immediate death.
Claudius Do it, England, / For like the hectic in my blood he rages, / And thou must cure me.

Act 4 Scene 6 - Two bereaved sons return

Polonius's son Laertes has returned to Denmark, determined to revenge his father's murder and blaming Claudius. Hamlet escapes his banishment and also returns. Already distraught, Laertes also learns that his sister Ophelia, driven mad by Hamlet's rejection and her father's murder, has drowned.
Laertes Let come what comes, only I'll be revenged / Most throughly for my father

Act 5 Scene 2 - The tragic climax

Unaware that Claudius has conspired to have Laertes murder Hamlet during a staged duel and for it to appear accidental, Hamlet accepts Laertes's challenge in good faith. As they fight, Gertrude drinks the poisoned wine that Claudius had intended for Hamlet and dies. Laertes manages to wound Hamlet slightly with the poisoned tip of his sword. In the heat of the fight, the swords are switched. Laertes is in turn wounded by Hamlet, using the poisoned weapon, and dies. Hamlet dies in Horatio's arms. Fortinbras arrives at the head of a Norwegian army.
Horatio Now cracks a noble heart. Goodnight, sweet prince/And flights of angels sing thee to thy rest