I think Shakespeare wants the audience to believe this is love at first sight. After all, in the first lines of the play, they are referred to as the "star-crossed lovers," meaning they were fated to meet and fall helplessly in love. And the action he creates transpires over a handful of days; at the end of these few days, Romeo and Juliet are willing to die for each other. Thus, there does seem to be some credence for the "love at first sight" analysis.
Romeo certainly proclaims his love for Juliet as soon as he beholds her:
Did my heart love till now? Forswear it, sight!
For I ne’er saw true beauty till this night. (I.v.55-56)
He immediately forgets his misery over Rosaline, which has plagued him for the entire play thus far, and becomes much more hopeful with this immediate change in demeanor. When he finds out that Juliet is a Capulet, he bemoans, "My life is my foe’s debt" (I.v.129).
Juliet is willing to be kissed by Romeo, teasing him playfully and inviting further affection. When he approaches her at her balcony just after the party, he hears her reflecting on their seemingly impossible predicament:
Deny thy father and refuse thy name.
Or, if thou wilt not, be but sworn my love,
And I’ll no longer be a Capulet. (II.ii.36-38)
Already, she is willing to give up her entire family if Romeo swears that he loves her. That's a fairly convincing statement.
Is it all effective? I think so. It's easy to fall into this whirlwind romance, hoping against hope that this will end well for this hopelessly-in-love young couple—yet knowing that it won't based on the predictions in the prologue. We cheer them on and then feel their collective pain at all the wrong turns in their plot, such as Tybalt's unfortunate death and Romeo's failure to receive that much-needed letter about Juliet's death not actually being real. I think the audience wants it to somehow all work out for them, meaning that Shakespeare does construct an effective love-at-first-sight portrayal.
Is it believable? Probably not. After all, Juliet isn't yet fourteen, and there are precious few people willing to get married after knowing each other for four days. But we suspend our sense of a believable situation all the time in the name of entertainment. Part of what makes a great story (whether we read it or watch it) is that it provides a wonderful escape from the realities of life. Their story may not be very believable, but Shakespeare creates characters whom we care about and a conflict that is still relevant over 400 years later.