Hamlet is upset by the ghost's tale, but agrees to do its bidding.
I am pretty sure that if you saw your father's ghost, you would be a little scared too. It is just not a regular thing. Even though the sentries and Horatio try to talk to the ghost, their reaction and Hamlet's reaction tells us that ghost sightings are not an everyday occurrence.
Hamlet is already a little off-kilter from returning home and his father's death, and especially his mother's wedding to his uncle. If you add that to the really strange occurrence of talking to his father's ghost, you can see where he would be a little panicky. Yet he goes with his father's ghost, and listens to his tale of woe. He pities the ghost, and when he hears it, he gets really upset. He seems to be almost having a heart attack!
O all you host of heaven! O earth! what else?
And shall I couple hell? O, fie! Hold, hold, my heart;
And you, my sinews, grow not instant old,
But bear me stiffly up. Remember thee!
Ay, thou poor ghost, while memory holds a seat
In this distracted globe. Remember thee! (Act 1, Scene 5)
Okay, so maybe Hamlet is a touch on the over-dramatic side. It is his father's ghost we are talking about here! Hamlet cannot believe what he is hearing as he learns that his uncle murdered his father. He is asking his muscles to bear his weight, because he is surprised and angry. He is so angry, in fact, that he decides to record his uncle's treachery.
O villain, villain, smiling, damned villain!
My tables,--meet it is I set it down,
That one may smile, and smile, and be a villain;
At least I'm sure it may be so in Denmark ... (Act 1, Scene 5)
I am not sure why he has to write it down. You would not think you would forget a thing like that! What Hamlet means is that he has to watch out for Claudius, because he is not what he seems. He is pretending to be Hamlet's friend, when really he is a villain! Nonetheless, Hamlet has had a rough night. His father's ghost has come and told him something terrible. Now he has to get his revenge.
Hamlet is not really the go-getter type. He is more of the brooding or scholarly type. He likes to think before he acts, and concoct over-elaborate plans. His father's ghost has given him a mission, and it is a complicated one. He has to avenge his father's death (but not take revenge on his mother). Poor Hamlet! What's a prince to do?