In Oedipus Rex, how does Oedipus learn his fate?
You have asked more than one question so I have edited it down to just one question. It is at the end of Scene 4 when Oedipus finally uncovers the truth that has been hinted at for so long. Oedipus finds a Shepherd who was born into slavery and served the old king Laius for all his life (the father of Oedipus). The Messenger who is present during this scene reminds the Shepherd that he gave him a baby boy to bring up as his own. The Messenger states that it was Oedipus who was that boy. The Shepherd tries to evade further questioning and confirmation to the best of his ability, but at last he is forced to admit that he was given a child from the house of Laius:
If you must be told, then...
They said it was Laius' child;
But it is your wife who can tell you about that.
It is the Shepherd who reveals to Oedipus the prophecies that had been given concerning him, that he would kill his own father. Ironically, the Shepherd explains that he pitied the baby and therefore gave him to the messenger to have a new life, never thinking that by so doing he would actually damn him. Note how Oedipus responds at this critical moment of self knowledge:
It was true!
All the prophecies!
O Light, may I look on you for the last time!
Oedipus, damned in his birth, in his marriage damned,
Damned in the blood he shed with his own hand!
It is then that Oedipus exits the stage to blind himself, to symbolically prevent the sight of the knowledge that he has just discovered.