One way in which Shakespeare makes the reader/audience believe that Romeo and Juliet have fallen in love is through the dialogue they exchange the moment they meet. There are two reasons why this dialogue is important. For one thing, it's playful, and people are more likely to fall in love while having fun. But for another thing, it also refers to religious motifs, and we know that religion is important to Juliet. Hence, the reader/viewer can easily see that Romeo and Juliet are connecting on a mental and spiritual level. We especially see the playfulness and flirtatiousness of their exchange when Juliet refuses to let Romeo consider touching her hand to be a defilement of her as a "holy shrine." Instead, she invites him to touch her hand, saying that pilgrims touch saints in holy devotion, as we see in her lines, "For saints have hands that pilgrims' hands do touch, / And palm to palm is holy palmers' kiss" (I.v.104-105). Not only is this a playful line, it shows her acceptance of him, which also hints that she will fall in love with him--Romeo will be as important to her as her religion, maybe even more important.
A second place in which Shakespeare makes the reader/audience believe that Romeo and Juliet are in love is in the famous balcony scene. In particular, Romeo swears by the moon that he loves Juliet, as we see in his lines, "Lady, by yonder blessed moon I swear ..." (II.ii.111-12). In addition, when Romeo asks for Juliet to exchange vows of love with him, Juliet's reply is, "I gave thee mine before thou didst request it," which again shows us that she is in love with Romeo (134).