Assuming Oedipus Rex is a quest, based on Foster's book How to Read Literature Like a Professor, how would Oedipus fit the quest pattern? Foster says every quest has a quester, a place to go, a stated reason to go there, challenges and trials, and the real reason to go. The challenges and trials and real reason are really easy to fill. We know Oedipus is finding himself and discovering his past. The quester of course is Oedipus, but the problem comes when we go to a place to go and a stated reason to go there. I came up with two ideas.
IDEA 1: A place to go - Thebes; stated reason - run away from his family because the prophecy said he would kill his father and marry his mother.
IDEA 2: A place to go - find the murderer of Laius (I don't know if this fits here); stated reason - save Thebes from a plague.
Of course I'd have to develop the ideas more, but I am looking for the main concepts. Thanks in advance.
Although this assignment is a good way for your teacher to ensure that you have read both Oedipus Rex and the book by Foster, this is one fairly limited approach to reading the play, and not one used by leading classical scholars. The play itself does not have any geographical motion; at the opening of the play, Oedipus is in Thebes, and at the end of the play he is preparing to leave Thebes. The goal of Oedipus during the play is to find the murderer of Laius and to end the plague. He also is trying to avert the outcome of the prophecy -- but his travels occured before the opening of the play.
The journey hinted at in the ending of the play is the one Oedipus will undertake when he leaves Thebes, which is described in another play by Sophocles, Oedipus at Colonus. Exiled and blinded, he searches for a place to die, and in his death brings a great blessing to his final resting place. Actually, this works far better as an exemplar of the quasi-Jungian quest model.
Thus both your ideas are good; the issue is that the narrow straight jacket of this particular approach doesn't really fit this play. When a theory doesn't fit the evidence, it suggests that literature is far more diverse than can be readily explained by any one theoretical model.
There are many points you can make about "going to a place." There is movement in the play. However, you have to make one differentiation. There is the intentional quest and the unintentional quest.
Oedipus' intentional quest involves his coming to Thebes to avoid fulfilling the prophecy that he would marry his mother and kill his father. The irony, of course, is that he fulfills what he wants to avoid. Another intentional quest is his desire to find the murderer of Laius. His determination actually drives the whole play. Even when people tell him to stop, he persists with blind conviction. Little does he know that he is indicting himself in the process.
Now as for the unintentional quest. Without Oedipus knowing, there is another quest. It is to live out the prophecy by finding his true identity. This is what makes the Theban plays so amazing. The quest is unwitting. If we go back to the idea of the "place," we can say that Oedipus' journey is the journey of self-realization.
I like your second idea much better -- the play is a "quest" in the sense of being a series of questions, or a kind of mystery that Oedipus is solving. I think your paper will be more successful if you adopt a bit more of a metaphorical sense of the "place" he is going. The play employs unity of place (and is the Aristotilean model for dramatic unity of action, time, and place), so as the other answers suggest, you won't be able to use a typical spatial approach to "questing."