There are probably many answers to this question. I would say that the first important event for Hamlet is seeing his father's ghost and learning that his father was murdered. Secondly, the at the performance of "The Mousetrap", Hamlet rewritten version of the "Murder of Gonzago", Hamlet sees his uncle's horrified reaction to the play. Claudius' reaction is confirmation to Hamlet that Claudius is guilty of his father's death. The final important moment would have to be either the funeral he witnesses for Ophelia or just before he goes down to fight Laertes. In talking with Horatio, Hamlet seems to have come to terms with life and death when he says," If it be now,
'tis not to come; if it be not to come, it will be now; if it be not now, yet it will come: thereadiness is all: since no man has aught of what he
leaves, what is't to leave betimes?" He seems to have come to grips with his angry demons and is ready to let fate/heaven take its course.
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The soliloquies of Hamlet which are spoken when he is under intense mental pressure reveal his true character and personality. I would consider the following three to be very important:
1. ActI sc2. Hamlet's first soliloquy reveals his anguish at his mother's haste in marrying Claudio: "Fraility thy name is woman!" He is tormented by the fact that his mother of all the people would marry again so hastily.
2. ActI sc5. After Hamlet's meeting with the ghost, Hamlet the dutiful son decides to avenge his father's murder: "And thy commandment all alone shall live/Within the book and volume of my brain." He decides to concentrate all his time and energies to avenge his father's murder so that his father's soul will rest in peace.
3. ActIII sc3. The most famous of all of Hamlet's soliloquies which begins "To be or not to be" and which expresses most poignantly Hamlet's existentialist angst.