Before Romeo and Juliet even meet, there are many other conflicts revealed in the play. The initial conflict in Romeo and Juliet is a disagreement between the two families. Sampson and Gregory are talking when Abraham and Balthasar approach. The Capulet's prejudice against the Montague's leads them to insult Abraham and a fight breaks out.
These four men begin the fight because they are from opposite feuding households. Benvolio and Tybalt join in the fighting and soon the townspeople are involved as well. It is almost a riot by the time the Prince and his people are able to stop the fighting. The Prince attempts to stop future fighting (ie resolve the initial conflict) by threatening Lord Montague and Lord Capulet with death if they persist in the feud.