He goes against his parents' wishes and marries Juliet in secret. That's a pretty gutsy move. Marriage in those days was not approached in so flippant a way as many people today go into it--marriage was until DEATH do us part--not until someone better comes along or you grip the toothpaste in the center too much.
Romeo also waits until the last possible moment to leave Juliet's side after they have cosummated their marriage. She begs him to stay saying that it was not the lark but the nightingale that they hear singing, and it is not until after he responds with "OK, come on with the day and let Romeo die for Juliet wills it so" that she comes to her senses about what that means and sends him on his way. He is quite willing to die even that early in the play for his love of Juliet.
Romeo's opening scene, in which he is pining over his love for Rosalind, is the first indication of his passion. Despite Benvolio's attempt to cheer him up, Romeo insists that he has been forever injured by Rosalind's rejection. He confirms what we have already been told, that he has spent his nights awake and walking, unable to sleep because his love is too powerful.
Going into the Capulet's garden also proves that Romeo is passionate about love. He is the enemy of the Capulet's, and is likely to be killed if found there. The audience already knows that Tybalt wishes to attack him for having attended the party. Juliet warns him away at first, fearing for his life. But Romeo refuses to go. He is too eager to first see, and then speak, with his new love to care about the danger he is putting himself in.
I would argue that there is another event that shows Romeo's passion, although in a very different way. When confronted by Tybalt, Romeo refuses to fight because he knows that Tybalt is the cousin of his wife. Despite all Tybalt's inflammatory words, Romeo will not be angered into drawing his sword. However, when Tybalt kills his good friend Mercutio, Romeo instantly goes on the attack. The love he has for his friend pushes him into the action he had been so steadfastly avoiding.