After meeting Juliet, Romeo's biggest transformation is that he changes from being depressed and sorrowful to joyful. Otherwise, Romeo's character remains the same. Romeo remains a rash, emotionally driven, impetuous youth.
When we first meet Romeo, he is emotionally torturing himself because he is in love with Rosaline who does not return his affection. He is reported by his father as being seen dawn after dawn crying under the same tree in the same part of town, presumably the side of town Rosaline lives in (I.i.127-129). Also, Romeo moans to Benvolio about how painful and heartbreaking love is, referring to love as a bunch of oxymora, particularly as both a hateful and an affectionate feeling, as we see in the line, "O brawling love! O loving hate!" (174). However, after having met Juliet, when we next see Romeo with Friar Laurence, Romeo is absolutely jubilant. He is so jubilant that Friar Laurence is shocked, declaring:
Holy Saint Francis, what a change is here!
Is Rosaline, whom thou didst love so dear,
So soon forsaken? (II.iii.65-67)
Hence, the only real way that we see Romeo change after meeting Juliet is that he changes from bestowing his affection on Rosaline to Juliet, thus changing from sad and depressed to joyful. Romeo is still being just as emotionally driven and rash, which is portrayed in his decision that he and Juliet absolutely must be married that day. Friar Laurence even questions the sincerity of his love for Juliet, saying that young men only love with their eyes and not with their hearts (68-69). He also cautions that the decision is rash, saying, "Wisely, and slow. They stumble that run fast" (97).
Hence, we see that in character Romeo remains the same rash, emotionally driven, impetuous youth after meeting Juliet that he is at the beginning of the play. The only difference is that he is now elated whereas before he was desolate.