There are a lot of online quote collections for Shakespeare. The plays are also all available online in the form of etexts. It helps if you have a particular play or topic in mind, but if not you can just type into a search engine "Shakespeare quote death" and so on.
My students always get hooked on these two lines in Hamlet:
"Frailty, thy name is woman!" (loved by the boys, of course)
"What a piece of works is man." (obviously the girls' choice)
William Faulkner must have thought this one was a really good since he found perhaps his most famous title in it:
Out, out, brief candle!
Life's but a walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more. It is a tale
Told by an idiot, full of sound and fury
Very memorable phrase, "sound and fury," in reading both Shakespeare and Faulkner.
That first quote, "He who has injured thee..." isn't Shakespeare, I don't think. It's Seneca.
Well, I don't know about 'best'. What do you mean? Do you mean 'the best things Shakespeare ever wrote'? (if you do, then it's completely down to your own - or my own - opinion: there'll be very little consensus!). If, though, you mean the most famous quotations from Shakespeare, there are hundreds.
Hamlet asks "To be or not to be? That is the question", before he later exclaims "Alas, poor Yorick. I knew him, Horatio". Juliet, frustrated, is forced to wonder of her new love, 'O Romeo, Romeo! Wherefore art thou Romeo?' Henry V begins with the famous chorus "O for a muse of fire...", and eventually commands his men to go "Once more unto the breach, dear friends, once more". Macbeth asks 'Is this a dagger that I see before me?'
And on top of those big ones, there are hundreds of smaller quotes that are just as famous. I love this little paragraph, written by Bernard Levin, which'll give you a few ideas:
On Quoting Shakespeare
If you cannot understand my argument, and declare ``It's Greek to me'', you are quoting Shakespeare; if you claim to be more sinned against than sinning, you are quoting Shakespeare; if you recall your salad days, you are quoting Shakespeare; if you act more in sorrow than in anger; if your wish is farther to the thought; if your lost property has vanished into thin air, you are quoting Shakespeare; if you have ever refused to budge an inch or suffered from green-eyed jealousy, if you have played fast and loose, if you have been tongue-tied, a tower of strength, hoodwinked or in a pickle, if you have knitted your brows, made a virtue of necessity, insisted on fair play, slept not one wink, stood on ceremony, danced attendance (on your lord and master), laughed yourself into stitches, had short shrift, cold comfort or too much of a good thing, if you have seen better days or lived in a fool's paradise -why, be that as it may, the more fool you , for it is a foregone conclusion that you are (as good luck would have it) quoting Shakespeare; if you think it is early days and clear out bag and baggage, if you think it is high time and that that is the long and short of it, if you believe that the game is up and that truth will out even if it involves your own flesh and blood, if you lie low till the crack of doom because you suspect foul play, if you have your teeth set on edge (at one fell swoop) without rhyme or reason, then - to give the devil his due - if the truth were known (for surely you have a tongue in your head) you are quoting Shakespeare; even if you bid me good riddance and send me packing, if you wish I was dead as a door-nail, if you think I am an eyesore, a laughing stock, the devil incarnate, a stony-hearted villain, bloody-minded or a blinking idiot, then - by Jove! O Lord! Tut tut! For goodness' sake! What the dickens! But me no buts! - it is all one to me, for you are quoting Shakespeare.
"This is very subjective. Different quotes appeal to different people at different times of their lives. I think the key is to really understanding the text at your own leisure with the help of a good teacher or a guide like Shmoop. Here’s a quote from Romeo and Juliet that I really like:
O fortune, fortune! all men call thee fickle:
If thou art fickle, what dost thou with him.
That is renown'd for faith? Be fickle, fortune;
For then, I hope, thou wilt not keep him long,
But send him back.
I consulted various Shakespeare sources that Shakespeare was the original source of the quote.
He who has injured thee was either stronger or weaker than thee. If weaker, spare him; if stronger, spare thyself.
And say to all the world, THIS WAS A MAN!
How poor are they who have not patience! What wound did ever heal but by degrees.
How use doth breed a habit in a man.
I am not bound to please thee with my answers.