Horatio is well-educated, & his opinion is trusted among the soldiers. When the ghost first appears, Horatio is skeptical, which is exactly why Marcellus asks him to come along:
Horatio says 'tis but our fantasy,
And will not let belief take hold of him
Touching this dreaded sight, twice seen of us.
Therefore I have entreated him along
With us to watch the minutes of this night,
That if again this apparition come
He may approve our eyes and speak to it.
So it's obvious that the other men hold him in high regard. There seems to be good reason for this too. Horatio is perhaps the character most concerned about Hamlet for Hamlet's sake, and not for any other ulterior motives. He honestly cares for his friend, a trait the Hamlet will acknowledge in Act III. Here, early on, when the ghost comes, Horatio warns Hamlet not to follow it:
What if it tempt you toward the flood, my lord,
Or to the dreadful summit of the cliff
That beetles o'er his base into the sea,
And there assume some other horrible form,
Which might deprive your sovereignty of reason
And draw you into madness? Think of it.
The very place puts toys of desperation,
Without more motive, into every brain
That looks so many fathoms to the sea
And hears it roar beneath.
Thus he warns Hamlet of the possible dangers of following what may well be a malevolent spirit.
Polonius is fat, annoying, and likes to speak in puns. Yet he also cares for his children, and offers advice to Laertes when he goes to leave:
There, my blessing with thee.
And these few precepts in thy memory
See thou character. Give thy thoughts no tongue,
Nor any unproportion'd thought his act.
Be thou familiar, but by no means vulgar.
Of course, it's good advice, but Polonius probably thinks he's the first one to think of it. Also, he's not just a fat jolly guy. He's got a rather sinister side, as shown when he convinces Reynaldo to spy on Laertes, even suggesting that he tells lies in order to find out Laertes' true reputation. Not the kind of thing most fathers do.
Hamlet is depressed, and perhaps even suicidal. He reveals this in his soliloquy, saying:
O, that this too too sullied flesh would melt,
Thaw and resolve itself into a dew,
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter! O God! God!
He establishes himself as melancholy from the beginning, due to his father's death and his mother's marriage. He will continue to identify himself by his melancholy throughout the play.
Gertrude notices this as well. She is worried for Hamlet, probably because she doesn't want Claudius to have a reason to suspect Hamlet of anything but grief over his father's death. Yet Gertrude senses Hamlet's disgust over their marriage, and she wants to make sure she's the only one. She gently chides Hamlet for his mood:
Good Hamlet, cast thy nighted color off,
And let thine eye look like a friend on Denmark.
Do not for ever with thy vailed lids
Seek for thy noble father in the dust.
Thou know'st 'tis common. All that lives must die,
Passing through nature to eternity.
The line that gives it away is "look like a friend on Denmark." She is specifically telling him to acknowledge the King as the royal head of state, and her husband as well. Of course, Hamlet refuses to do this, & this will determine his action (or lack thereof) throughout the play.