I've always liked the very first line of Hamlet, for it both introduces and epitomizes the confusion in Denmark and sets us up for the conflicts and chaos which ensues. The line is only two words long, but it is the context which make the line significant. The man on guard is Francisco; if someone approaches, it is he who should be asking who it is. Instead, it is Bernardo, the one who approaches the guard and asks "Who's there?" Things in Denmark are clearly in disarray, and this line represents that.
I love the poetry of these lines:
But look, the morn, in russet mantle clad,
Walks o'er the dew of yon high eastward hill.
Break we our watch up; and by my advice,
Let us impart what we have seen tonight
Unto young Hamlet; for, upon my life,
This spirit, dumb to us, will speak to him.
The poetry of those lines and the most important part of the scene are embodied in them.
I really like the description of Fortinbras as it is relayed by Horatio. I cannot see what his physical person looks like, but I can clearly imagine his attitude and how he probably carries himself. Horatio says that Fortinbras is
of unimproved mettle hot and full
[who] hath in the skirts of Norway, here and there,
Sharked up a list of lawless resolutes
For food and diet to some enterprise
That hath a stomach in't.
I can just imagine a brash young man who is hot on the idea of gaining back the lands lost by his father, and gaining them back at any cost. He isn't in command of the formal army of Norway -- he hired mercenaries who are willing to do whatever it takes to get the job done. He is of unimproved mettle meaning he is still raw, young, unseasoned, which I think makes him more dangerous. I also like the verb "sharked" -- it makes the his selection of "warriors" seem so indiscriminate -- part of that young energy that he displays. We so clearly understand what kind of person he is, it is clear how he is a foil to Hamlet as he presents himself in scene 2.
As Horatio, Marullus, and Bernardo confront the ghost in the dark on the battlements, the ghost is about to speak to them when a rooster crows. Hearing the sound, the ghost is startled and departs. Horatio explains the ghost's sudden disappearance:
I have heard
The cock, that is the trumpet to the morn,
Doth with his lofty and shrill-sounding throat
Awake the god of day, and at his warning,
Whether in sea or fire, in earth or air,
The extravagant and erring spirit hies
To his confine: and of the truth herein
This present object made probation.
It's good to know that we are safe from ghosts and "unconfined" spirits as soon as the sun comes up! One particular line appeals to me because of its poetic imagery, the reference to spirits wandering "in sea or fire, in earth or air." The whole passage is rich in mystery and supernaturalism, engaging our imagination. What really does "go bump" in the night?
You are obviously going to receive some very subjective responses to this question, as the reasons why anyone will like a quote will be very different for each person. However, for me, I often find myself muttering some of the lines of Hamlet's soliloquy in Act I scene 2 in my more existential moments. They seem to capture the bleakness of his situation and how life really has so little to offer for him. Note what he says:
How weary, stale, flat and unprofitable
Seems to me all the uses of this world!
Here he is, trying desperately to make senses of a very different kind of world and struggling to understand it and find his role in it and failing. I think there are times when all of us can relate with such changes and this quote expresses perfectly intense dissatisfaction with the world and our place in it.