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.