While some modern audiences like to scoff at the speed of Romeo and Juliet's romance, it must be remembered that Shakespeare is not exactly operating on a realistic model of time. Time is intensified and concentrated in the play to emphasize Romeo and Juliet's passion and youth. As for the question of whether or not love at first sight is realisitc, in real life there are people who claim to have fallen for their partners right away. It is rare but not unheard of at all.
Whether or not their love is "real" of course is all speculation. Undoubtedly, it is initially based on physical attraction: when Romeo sees Juliet dancing at the ball, he comments on her beauty, not her personality. Juliet too experiences a wave of desire for Romeo. However, to call their feelings for each other lust alone is not quite accurate. Lust goes only as far as the bedroom. Romeo and Juliet clearly have plans to stay together for a lifetime.
After their marriage, Romeo and Juliet show a willingness to sacrifice things for one another, most significantly their social lives in Verona. It must be remembered that both of them are young; they are still very much attached to their families. They are willing to sever ties with their parents and their home town in order to be with each other. That's why when Juliet claims she will "no longer be a Capulet," her statement is more significant than one might initially realize: she is more than willing to destroy her old identity to forge a new one with Romeo by her side.
Of course, some have often wondered if the relationship would have "lasted." The two are never married long enough to experience real marital difficulty, disagreements, or regret over their hasty decisions. However, this ignores the real tragedy of the story: that Romeo and Juliet never get that chance because the feud sees fit to ensure their lives are cut short. So it must be assumed on the evidence within the text that Romeo and Juliet do at the very least care about each other in a way that goes beyond teenage hormones.
Romeo and Juliet fall head over heels in love with each other the moment they lay eyes on each other at the Capulet party. This is by no means an uncommon occurrence, and there are many people with long and happy relationships behind them who fell in love like the "star-cross'd" lovers.
At the same time, it's only later on that we discover the true nature of the love that Romeo and Juliet have for each other. At first, it would seem that Romeo is just infatuated with Juliet, in much the same way that he was infatuated with Rosaline. For her part, Juliet is certainly attracted to Romeo, but it's by no means immediately apparent that her love is anything more than a girlish crush.
In due course, however, Romeo and Juliet prove that their love is real by word and deed. And it's the deed that's most important here. Romeo and Juliet show their love for each other by defying the express wishes of their feuding families and getting married. This is a big commitment, one that they take freely and without external pressure. At no point do either of them give the impression that they think that getting hitched is a bad idea.
While one cannot read the contents of someone's heart, it does seem reasonable to conclude that, by their defiance of their feuding families, and their commitment to each other through the holy sacrament of marriage, Romeo and Juliet are showing us that the love they share is very much the real thing.
Falling in love at first sight is indeed a believable situation; however, it's doubtful that it's real love. It is more a feeling of infatuation, which is a "foolish or all-absorbing passion," rather than real love (Random House Dictionary). Real love is a matter of choice rather than just a feeling. Real love is felt when you truly get to know the person and decide to accept the good with the bad. What Romeo and Juliet felt when they first met, which they deemed as love at first sight, was really more of an intense, passionate physical attraction. However, as the play progresses, Juliet does mature enough to feel real love for Romeo.
Romeo's love at first sight is most clearly seen as being infatuation the moment he lays eyes on her. Romeo very clearly mistakes real love for physical attraction; however, it's perfectly believable that Romeo thought he felt real love at first sight, as young men frequently mistake love for physical attraction. We especially see that Romeo has mistaken love for physical attraction when he says, "Did my heart love till now? Forswear it, sight! / For I ne'er saw true beauty till this night" (I.v.54-55). But since Romeo clearly believes he has fallen in love, Shakespeare portrays falling in love as a believable situation.
While Juliet clearly felt equal physical attraction for Romeo, her love does progress to real love the moment she learns that Romeo has killed her cousin Tybalt. Juliet is faced with making some difficult decisions, typical of the decisions people who are in a real-love relationship must face: (1) Should she hate him or continue to love him? (2) Should she distrust him or continue to trust him? At first, her negative emotions prevail, and she feels very deceived by him, as we see in her string of oxymora:
O serpent heart, hid with a flow'ring face!
Did ever dragon keep so fair a cave?
Beautiful tyrant! Fiend angelical!
Just opposite to what thou justly seem'st--. (III.ii.76-81)
These oxymora make it very clear that Juliet used Romeo's handsome looks to judge his character, making herself believe she was really in love with him when in actuality she knew little about him. Now that it seems he has done a treacherous deed, she is faced with the possibility that he is not what he seemed to be. For example, she describes him as both a serpent and a person with a "flow'ring face." The serpent is typically seen as a symbol of evil and sin, and being "hid with a flowr'ing face" refers to his beauty. Therefore, she is arguing that his beauty deceived her, and his face actually hides an evil heart. However, Juliet does not continue to see him as deceptive. As the scene progresses, she reasons herself into seeing that Romeo must have killed Tybalt out of self-defense. She further decides that since he is her husband, she must continue to trust and honor him, showing us that the infatuation she felt for Romeo at the beginning has now blossomed into real love.