One way in which we see fate operating in Act 1, Scene 1 is that we learn Romeo has been rejected by Rosaline. Rosaline does not have any known genuine reason for rejecting Romeo. He is one of the handsomest men in Verona, as we see Juliet's nurse point out in the lines, "Though his face be better than any man's ..." (II.v.41). Plus, he is reasonably wealthy, but Rosaline will not even be moved by that, as Romeo points out in the line, "Nor ope her lap to saint-seducing gold" (I.i.216) In this line, the word "ope" can be translated as "open." In other words, Romeo is saying that Rosaline will not even respond sexually to the fact that Romeo has "gold," or money. Since Rosaline does not have any known reasons for rejecting Romeo, we can say that her rejection is an operation of fate.
However, while fate may govern Rosaline's rejection of Romeo, he has made the choice to respond to the rejection as he is doing. He has made the choice to let her rejection make him completely miserable to the point that his friends are concerned about his sanity.
One way in which we see fate play out in Act 1, Scene 2 is that we see Capulet's servant show Romeo the list of guests that Lord Capulet is inviting to his feast that night. Among the names that Romeo reads is his friend Mercutio, as we see in the line "Mercutio and his brother Valentine" (I.ii.70). It seems that Capulet's servant has stopped to ask Romeo to read the list because the servant cannot read himself. However, it was fate that the servant should happen to stumble on Romeo rather than anyone else in the city.
While it was through fate that Romeo learned of the feast and that both Rosaline and his friend Mercutio will be attending, it was Romeo's choice to attend the ball. He even made the choice against his own better judgement, considering that he had a dream the night before that he would die as a result of some adventure, which he took as an omen.
In Elizabethan England, astronomy was taken very seriously, and was often used as a deciding factor in romantic and business decisions. The statement that Romeo and Juliet are "star-crossed" refers to the clashing of the Zodiac signs and how this will negatively affect their future together. Therefore, the fight between servants and, later, the noble houses themselves foreshadows the ill (and, on one side, violent) end to which our two lovers will come. Additionally, in Act One, Scene II, Juliet expresses her ignorance of romantic relationships and Romeo can't get Rosaline off his mind. This shows that fate is working against the pair preparing mentally and emotionally for romance leading to commitment.