Marcellus sees the ghost as a fearful apparition. He is afraid to do anything that might anger the ghost, but he seems to also have his doubts about the ghost's purpose. Marcellus is the one who asked Horatio to come with him and the other guards to see the ghost and attempt to speak to it. He knows that Horatio is an educated man who would be able to speak to the ghost. He thinks the ghost's appearance has to do with the war preparations that are going on in Denmark at this time. Marcellus asks Horatio, after the ghost appears to them, why Denmark is making such preparations. When the ghost makes a second appearance to them that night, Horatio then asks the ghost if his reason for appearing has to do with the war preparations. Marcellus believes that threatening the ghost is not beneficial, and later, after the ghost appears to Hamlet as well, Marcellus is the one who makes the comment that something is "rotten in the state of Denmark" meaning that again, he thinks the ghost's appearance has something to do with what is to come.
Horatio is a doubter. When he first comes into the scene in Act 1, sc. 1, Marcellus tells Bernardo that Horatio says it is just their imaginations; there is no ghost. Horatio is a stoic, one who tries to use logic only, and a ghost defies logic, so Horatio doesn't believe in ghosts until he, too, sees the ghost of King Hamlet. Then he is astounded and does not know why the ghost appears. He asks it if its reason for appearing has anything to do with Denmark's war preparations. He begs the ghost to speak to him and tell him. He fears that the ghost may not be a benevolent figure when he warns Hamlet in Act 1, sc. 4, to be wary of the ghost. It is Horatio who decides taht Hamlet must be notified and brought to the ghost at the end of Act 1, sc. 1. Hamlet is much more the romantic than his friend, Horatio, so when he is told of the ghost's appearance, he is quick to believe.
Hamlet is not afraid of the ghost and does not believe that the ghost would lead him to harm, so when Horatio warns him to be careful, he is not worried. Later, Hamlet expresses some doubt about the ghost, about whether or not the ghost is the true spirit of his father or possibly a devil sent to bring him to hell (Act 2, sc. 2). This is the reason Hamlet decides to have the players perform a scene depicting the killing of King Hamlet as related to Hamlet by the ghost. After the king's reaction to the play, Hamlet is sure the ghost is indeed, the spirit of his dead father.