Hamlet makes two major choices that lead to his demise as well as the downfall of others:
Believing the Ghost: When Hamlet makes the choice to listen and believe the apparition of his dead father, he willingly buys into the spirit's claim that he has been murdered by Claudius:
Bring with thee airs from heaven or blasts from hell,
Be thy intents wicked or charitable,(45)
Thou comest in such a questionable shape
That I will speak to thee. (I.iv.44-47)
This moment of decision has huge repercussions for the rest of the play, influencing Hamlet's decision to investigate his uncle.
Pretending to be insane: Hamlet's choice to feign insanity leads to major issues in the play; after speaking with the ghost, he cautions Horatio and Marcellus not to say anything about his off-the-wall behavior:
How strange or odd soe'er I bear myself—
As I perchance hereafter shall think meet
To put an antic disposition on— (I.v. 190-192)
Hamlet's disruptive behavior may be his strategy to deflect attention away from his search for the truth about his father's death, but his mad behavior strains his relationship with Ophelia, and ultimately he sends her so many mixed signals that she really does go insane.