How does Oedipus develop and change throughout Oedipus Rex?
It depends how you read the play, of course. It's a great play, this one, and so it sort of pushes back at simple questions. I'll show you what I mean.
At the start of the play, Oedipus has it great. He's King of Thebes, and he's 'determined' (interesting word for him to choose) that he's going to rid the city of the plague that is killing its people. An oracle tells the Thebans that a murderer is in the city and has to be rooted out and removed - then the plague will go away. And Oedipus is determined to find the murderer and kick him out. Oedipus is single-minded, some might say obsessive, but - you know what? - he's a good king. That's what you want, right?
And even though, throughout the play, Tieresias, Jocasta, and various other characters tell Oedipus to stop, to turn back, to focus on something else, to stop searching for the murderer, he won't. He downright refuses to stop looking.
And when, at the end, he realises that the murderer is - in fact - him: Oedipus himself, who has, unknowingly, murdered his father - he still doesn't change. Still single-minded, still determined, he blinds himself and then kicks himself out of Thebes. He's sort of impressive and scary at the same time (the Greek 'deinos' means both terrible an dwonderful - and it's a good word for Oedipus).
So in that sense, he doesn't develop or change. He's like a bulldozer plowing through the play, smashing apart resistance until he finally comes to smash apart himself.
In another sense, he changes profoundly. At the start of the play he's the son of (he thinks) the King and Queen of Corinth, married to the beautiful Jocasta, his wife (with whom he has children). By the end of it, he's the son of the King of Thebes, Laius, whom he murdered, and both husband and son of Jocasta (who he married without knowing she was his mother). So if you're talking about his familial roles, well... he changes pretty totally. But personally, I'd say one of the interesting things about the play is that he doesn't really change at all personally. And the man looking for the murderer does indeed get his wish, and find that murderer out by the end. It's just a shame that the person he ends up catching with the blood on their hands is himself.
Oedipus moves from hubris or pride to humility and self-knowledge over the course of the play. At the start of the play, it is completely incomprehensible to him that he could be responsible for the plague devastating Thebes. He is blind to his flaws and his fate. He thinks he is a good, upright person, a fine husband and father, and an able ruler. As the action of the play unfolds, he comes to realize that he is the root cause of the disease in Thebes. Abandoned at childbirth and left to die, Oedipus unknowingly killed his father on the road to Thebes years ago. Then he married the queen, his father's widow, having no idea she was his mother. As he realizes what he has done, his concept of self alters. He goes from thinking himself unblemished to being in so much pain he voluntarily blinds himself. Ironically, this event marks his first true insight about himself. He knows now he is imperfect, and this knowledge lends him wisdom and humility.