Act I Scene I begins with the mention of the "thing" -- that is, the apparition. Three of the people on the watch -- Bernardo, Francisco, Marcellus, -- do not believe that the "thing" might be anything other than a ghost. The belief in the appearance of spirits is unquestioned among them; it is taken as a point of fact that ghosts exist and appear to men. Only Horatio, the man of science (in the minority, it must be noted) is unable to believe in a ghost. Horatio is brought along, in fact, to be made to believe in the existence of the ghost, not the other way around (to disprove its existence.) When the Ghost appears, of course, all are horrified. But Horatio fights against superstition, and asks the Ghost what it is. Horatio does try to find a "logical explanation" for the appearance of a figure so like the dead king. But even Horatio, the scholar, jumps to the conclusion that this can only be an evil omen "This bodes some strange eruption to our state" (I.i.83)
The King Claudius, in the second scene, immediately uses the word "green" (in this case meaning fresh, but also a clue to the reader.) Claudius was, indeed, green with envy of his brother, and killed him for his kingdom and his queen. Hamlet refers to this envy in many of his retorts back to Claudius. When Horatio tells Hamlet of the appearance of the Ghost, Hamlet seems inclined to believe in the existence of ghosts, and, since the Ghost is armed, Hamlet also believes it is an evil portent.
The third scene consists of conversations between the brother and sister Laertes and Ophelia and their father Polonius, and they are concerned more with common sense in their relationships with others than with envy or superstition. But in Scene 4 the Ghost appears to Hamlet and Horatio. Hamlet instantly believes it to be the ghost of his father, and calls on "Angels and ministers of grace" to defend him (line 42). He doesn't think that it could be (for, in the play it is not) someone pretending to be a ghost to frighten him. He is willing to believe in the supernatural, and calls upon the guardians of heaven, not his own armed watchmen, to defend him from it. Of course Horatio and Marcellus are frightened of it, and warn Hamlet from going to speak with it privately when it beckons him. They imagine that it might lead him into dangerous or deadly actions. Again, as the scene ends, the men are convinced that the appearance of this apparition can only lead to bad things for the kingdom of Denmark.