In Act III scene 1, why is Tybalt looking for Romeo?
Initially, there is no evidence in this particular scene that Tybalt is looking for Romeo. He does not inquire about the young Montague's whereabouts when he comes upon Romeo's friends, Mercutio and Benvolio. All that he does is tell Mercutio, "...thou consort'st with Romeo." He is saying that Mercutio is associating with Romeo and stating this as an accusation—as if there is something wrong about being friends with Romeo.
We already know, at this point, that Tybalt has lodged a written challenge with the House of Montague in which he states that he wishes to duel with Romeo in defense of his honor. Tybalt felt insulted in Act 1, Scene 5 when Romeo and his companions gatecrashed the Capulet Ball. The occasion was supposed to be a private affair. When Tybalt wanted to challenge Romeo at the time, he was humiliated by his uncle, Lord Capulet, who angrily admonished him and told him to leave Romeo alone. Tybalt then swore revenge:
I will withdraw: but this intrusion shall
Now seeming sweet convert to bitter gall.
The implication is that Romeo's pleasure (for attending the party and getting away with his infringement) will soon become a regrettable action.
In Act 3, Scene 1, Tybalt is clearly looking to make trouble, and he obviously knows that Mercutio, just like him, has a quick temper. It will be easy to provoke him into a fight. As it is, his insult has the desired effect, and Mercutio responds by stating that he is offended by Tybalt's suggestion that he and his friends are members of a band of musicians because they "consort." He threatens Mercutio and states that he will use his fiddlestick—his sword—to make Tybalt dance.
Romeo arrives on the scene, and Tybalt quickly turns to him. It becomes evident that Tybalt has been trying to find Romeo when he states, "Here comes my man." Tybalt's potent dislike for Romeo is displayed when he tells him:
Romeo, the hate I bear thee can afford
No better term than this,—thou art a villain.
Romeo, however, has fallen in love with Tybalt's cousin Juliet and secretly married her, and he sees no reason to fight. He regards Tybalt as one of his family because of his association with Juliet. He dismisses Tybalt's challenge and tells him that he has never injured him and loves him more than he can ever know. He tells Tybalt to be satisfied with what he says.
Mercutio is shocked at what he believes is Romeo's cowardly response. He draws his sword and asks Tybalt to fight. The well-meaning Romeo intervenes, but, unfortunately, his action gives Tybalt the advantage, and Tybalt stabs Mercutio from under Romeo's arm. Mercutio perishes from his injury.
The incident sets off a series of events which eventually culminate in the tragic and unnecessary deaths of many.
In scene 1 of act 3, Tybalt is looking around the city of Verona for Romeo. In Act I Romeo and his friends crashed the Capulet's party. While we know they did this in an attempt to cheer up Romeo, Tybalt is convinced this was an act of aggression and believes that his family has been insulted. Since Lord Capulet prevented Tybalt from settling the score at the party, he plans to take care of things in the light of day. When he sees Romeo Tybalt says:
Romeo, the love I bear thee can afford
No better term than this: thou art a villain.
Even though Romeo, who has just married Tybalt's cousin, tries to explain why he does not dislike Tybalt, but in fact loves him the way he would anyone in his family, Tybalt will not listen to him.
Boy, this shall not excuse the injuries
That thou hast done me; therefore turn and draw.