Romeo changes in a few key places. When the audience first meets Romeo, he is a love sick, whiny teenager. He's simply lost in grief that Rosaline does not reciprocate his love for her. It's actually really funny. She would rather be a nun than date Romeo. Seriously.
Then she hath sworn that she will still live chaste?
A few scenes later, Romeo is hopelessly in love with Juliet. He's smitten with her beauty and grace, and she becomes the next target of his all consuming love. Because Romeo is now in love with a Capulet, he no longer sees the point in fighting the Capulets. That's why Romeo refuses to fight Tybalt. Romeo's a lover, not a fighter.
Of course that all changes after Tybalt kills Mercutio. Then Romeo changes into a lover and a fighter.