What could be argued as the most famous of all Shakespearian quotes resides in Act III, the "to be or not to be" monologue. Look to scene 1, lines 56-89 for that speech. In it, Hamlet ponders whether life is worth living, and what keeps himself or others from just ending life. This is a major theme of "Hamlet"-his pondering of life's value, and death's meaning.
Then, Ophelia comes in and you get several quotes from Hamlet touching on the theme of his disgust with women. The "Get thee to a nunnery" quotes(1. 122, 139) show Hamlet's bitterness towards the fickle nature of women, an indirect hit at his mother, who he is very upset with. It is also a foreshadowing of the conversation he will have in his mother's bedroom in scene 4.
Look at the very end of scene 3 for foreshadowing of Hamlet's intentions to kill the king (he comes upon the king praying, but decides to wait until the king is doing something evil to kill him).
In scene 4, he rants and raves at his mother, telling her "I must be cruel, only to be kind" (l.177), a famous quote, hinting at his need to be cruelly "truthful" with her to help open her eyes to the situation as he sees it. He does this with many other characters also: Ophelia, Polonius, the king, his friends. He behaves cruelly and bizarrely in order to seek out the truth of his father's death.
Those are just a few examples, and I hope that it helps!