The role of free will and fate is one of the central themes in Oedipus Rex by Sophocles, and the argument begins even before the play opens. Because the play opens in the present and the past (antecedent action) is revealed slowly as the play unfolds, it is tricky to know exactly what "at the start" of the play means.
If we consider the literal beginning of the play, we see the forces of fate and free will first in the inability of Oedipus to act on his city's behalf. Things are so desperately awful in Thebes that the people have come to the temple and the palace to beg their king, Oedipus, to do something before everything and everyone dies. He obviously loves his people and has let things get so bad in his beloved city because he believes that he--or perhaps all of them--could eventually change things. Yet things just keep getting worse and Oedipus got desperate enough to consult the gods, an admission that he is not in control of what happens. All he knows to do has already been done: he has sent Creon to the oracle in order to find out what is cursing his city and how to fix the problems.
This simple act makes everyone happy, but it also reveals the mindset of both the people and the king. They all obviously believe that what is happening to them is out of their control, and therefore the solution is equally out of their control. This implies their strong belief in fate, which in their minds clearly supersedes their individual and collective free will. While no one says it explicitly, we must assume that the things they have done--the actions they have taken to alleviate the terrible issues--have not worked, and therefore they have determined that they are not free agents who can impact their own futures and destinies. Instead, it is fate which has done this to them, and it is fate which must change things.
This idea of fate superseding free will is reinforced when Creon and Teiresias arrive. Tiresias tells Oedipus the truth, but the king does not "see" it. Oedipus rather browbeats the man of the gods, and finally Teiresias vows to speak no more because he can see that Oedipus is stubborn and will not believe him. They have this exchange:
TEIRESIAS: Well, it will come what will, though I be mute.
OEDIPUS: Since come it must, thy duty is to tell me.
TEIRESIAS: I have no more to say; storm as thou willst,
And give the rein to all thy pent-up rage.
Everyone else believes what the prophet says, that there is no changing fate, but Oedipus still does not quite accept this and things get much worse for him because of it.
If we consider the antecedent action, that which happened before the action of the play begins, we do see some hint that free will might indeed be exercised--but it just a hint. When Oedipus received the prediction of the oracle that he would one day kill his father and marry his mother, Oedipus immediately takes his fate into his own hands (which is another way of saying he exercised his free will) and he runs away from his home and his parents. This is an example of free will which happens early in the play but of course is not revealed until much later; however, we know how this ended for Oedipus, and it seems that his actions were actually ordained by fate and lead him exactly to the situations he hoped to avoid.
In both cases, Oedipus seems to think he has free will but is eventually forced to realize that his will is subservient to fate.