There are many different reasons why Shakespearean drama can be seen as universal and timeless. Two reasons are themes and characterization.
One prominent theme in Romeo and Juliet is love vs. hatred. Love and hatred are both timeless emotions. All throughout the ages, people have been known to hold grudges and to prolong arguments, even turning arguments into "ancient grudge[s]," as the First Prologue refers to (3). A second prominent, timeless theme is the consequences of violent, passionate, uncontrolled emotions, whether the emotions are love or hatred. While the Capulets and Montagues are certainly guilty of feeling violent, uncontrolled hatred towards each other, leading to many deaths, Romeo and Juliet are also guilty of feeling rash, violent, passionate love, which contributes to their own untimely deaths. Friar Laurence classifies the situation best when he warns, "These violent delights have violent ends / And in their triumph die, like fire and powder" (II.vi.9-10).
However, while Romeo is guilty of rash, violent, passionate emotions, his characterization is also universal and timeless. All throughout the ages, young men have been guilty of being driven by their emotions rather than their rational minds. We see Romeo being guilty of thinking irrationally when he replies to Benvolio's wise advice to stop thinking of Rosaline, "O, teach me how I should forget to think!" (I.i.228). Not only that, all throughout the ages, young men have been guilty of confusing lust with true love, just as Romeo does with respect to both Rosaline and Juliet, again showing us just how much Romeo is a universal and timeless character.
If I understand what you are asking, some quotes that are univesally remembered are
"Romeo, Romeo, where for art thou, Romeo?"
"What is in a name, a rose by any other name would smell just as sweet..."