Good question. Juliet's age is, you're quite right, specified by the Nurse and Lady Capulet in Juliet's first scene as not-quite-fourteen.
As for Romeo's age, Shakespeare never tells us! We know - from what Friar Laurence and the Nurse comment on various occasions - that he is a "young" man (2.4.119) but we don't know precisely how young. A stage tradition has grown up of playing him as older than Juliet.
Shakespeare's sources don't really help either. Arthur Brooke's poem about Romeo and Juliet has Romeo's "tender chin" without a beard - suggesting, perhaps, anything from 12 - 18? And Bandello - whose novella Giulietta e Romeo was the source for Brooke's poem - gives his age as 20 or 21.
So the answer is - we just don't know.
Romeo and Juliet is the very famous story of the two "star-crossed lovers" (Prologue to Act I, line 6) who are doomed to fail in their attempts to overcome the feud which has dominated the Montague and Capulet families for as long as anyone can remember. They clearly underestimate the stubbornness of their parents and the hatred in their hearts which even extends to the servants who bear as much animosity towards the opposing household as they would towards a true enemy.
Romeo and Juliet's inability to grasp the true depth of feeling involved in this "grudge" (Prologue, line 3) gives an indication of just how young they both are. Neither Juliet nor Romeo appreciates the complexity of the problem and they even believe that denying their ancestry may be enough. Juliet ponders that "a rose by any other name would smell as sweet" (II.ii.44) suggesting that a family name should not have any bearing on her relationship with Romeo and she has come to the conclusion that it is "but thy name that is my enemy" (38). Romeo is also willing to renounce everything for her because as he says, his "name is hateful to myself" (55). Romeo is not yet old enough to understand that he would be expected to uphold the resentment and loathing in honor of his family name.
The ages of Romeo and Juliet are significant in understanding how they come to such a tragic end because it is their youth and inexperience and their inability to see beyond their immediate circumstances that causes them, Romeo in particular, to act so irrationally. Romeo is so overwhelmed by his circumstances and so immature in his actions that he is apparently just a teenager of perhaps sixteen years old, definitely younger than eighteen because by eighteen a young man of his standing would be expected to be able to lead men into battle and so he would not act so impulsively.
As robertwilliam states, Shakespeare never specifies Romeo's age. It is generally assumed he is a few years older than Juliet. The depiction of his essential youthfulness is what really matters in any case. He displays the ardent passion, the romantic idealism and also the rashness that one might say is natural to a youthful temperament.
It goes without saying that Romeo and Juliet have become the archetype of young, ill-fated lovers, attempting to breach the conventions of the adult world that surrounds them. The irony is, of course, that their elders and supposed betters display a lack of rationality and maturity in continuing their bitter feud, seemingly just for the sake of it. It is a pointless feud which ends in tragedy. Against such a backdrop, the youthful lovers appear to have better sense and judgement than their parents. This applies particularly to Juliet, who, as many have noted, displays a sense of dignity and shows a resourcefulness which would seem to belie her tender years. She is younger than Romeo but in some ways she appears the more mature of the two.
My teacher said he's about 16-17 so he's not as that old comparing to Juliet, who is 13, turning 14
romeo has been said to be anywhere from 18-23 however, regardless of which it is, he is still quite a bit older then Juliet who is only 13.
That's a good question, Romeo is supposedly to be around 18 or 19 years old.
Romeo is about 14/15 in the beginning of the book.
about 4-8 years older than juliet
I have always assumed he was 16