Barnardo, Marcellus, Francisco have been seeing a ghost at the castle and Horatio doesn't believe them. He stays until the ghost appears again and now he believes them, but now they think it looks like there last king Hamlet.
Horatio"Let's do't, I pray and I this morning know Where we shall find him most convenient" (1.i. line 189-190).
Hamlet grieves for his father and wants to leave Denmark but Gertrude, Claudius and the King convince Hamlet to stay. Hamlets mother quickly got married after her husbands death.
"Tis sweet and commendable in your nature, Hamlet, To give these mourning duties to your father"( 1.ii. line 90-93)
Polonius and Laertes are giving Ophelia advice and asking her questions about her relationship with Hamlet. That Hamlet is not in the right place right now to be in a relationship because of his grieving for his dad and mom marring again.
"You do not understand yourself so clearly As it behooves my daughter and your honor. What is between you? Give me up the truth" (1.iii. Line 105-107).
Marcellus and Horatio try to convince Hamlet to not go after the ghost, but Hamlet wants answers. Hamlet does not have a stable mined and he is not thinking about the dangers the ghost can bring him., by following it.
"What if it tempt you toward the flood. my lord? Or to the dreadful summit of the cliff..." (1.iv. Line 77-78).
The ghost talks to Hamlet and tells him how his father actually died and who did it. Hamlet then tells Horatio only some parts of what the ghost said. And if Hamlet gets caught then he can get in serous trouble about knowing what the ghost is telling him.
"Let us go in together, And still your fingers on your lips, I pray. The time id out of joint. O cursed spite That ever I was born to set it right! Nay, come, let's go together" (1.V.Line 208-212).