Romeo is often called impetuous, in that he acts often without thinking beforehand. However, even when he does stop to think about the consequences of his actions, he still steps in. He demonstrates bravery in this way.
When we meet Romeo he is lovesick after being denied by Rosaline. However when given the option of attending a Capulet ball, he takes it. He knows of the feud between the Capulets and the Montagues, and he knows that by going he could very well spark more fighting. When he is spied by Tybalt, only Capulet stops him from engaging in a fight with Romeo the gate-crasher to avoid retribution from the Prince.
He of course meets (and falls for) Juliet at the party, but demonstrates his bravery later. When he is told that she is a Capulet, he could have left her at that. However, he leaves his friends and scales the walls of the Capulet mansion, and the famous Balcony scene begins.
With love's light wings did I o'erperch these walls, (70)
For stony limits cannot hold love out,
And what love can do, that dares love attempt:
Therefore thy kinsmen are no stop to me.
He again demonstrates bravery when she asks of marriage. He risks being exposed fraternizing with the enemy by telling Friar Laurence of his plans to wed Juliet. Fortunately for him, the Friar sees this union as a chance to mend the feud for good. Romeo again risks his own self in being seen talking with Juliet's nurse about the wedding plans, and at the wedding itself.
When Tybalt kills Mercutio, Romeo quickly moves into action to avenge Mercutio's death. He slays Tybalt, his new wife's cousin, even knowing that the Prince will not stand for the death, and he will be sentenced to death himself. (Fortunately, Romeo is merely banished for his deed, given that Tybalt killed Mercutio first). Again his bravery shows through when he confesses this to the nurse, risking that Juliet will no longer love him.
Now, Tybalt, take the villain back again, That late thou gavest me; for Mercutio's soul Is but a little way above our heads, Staying for thine to keep him company:Either thou, or I, or both, must go with him.
While banished, Romeo hears of Juliet's death. This leads to a brave rush back to Verona, where he fights Paris, who had been guarding and mourning Juliet at the tomb. He kills himself, but his bravery in returning to the city is to be noted.