How does Duncan reward Macbeth for defeating the rebels and in what order does Macbeth discover this?

King Duncan rewards Macbeth's bravery with a promotion.

In Act I scene ii, a "bloody sergeant" tells King Duncan that Macbeth bravely slew Macdonwald, the leader of the rebels, by slicing him from navel to chin and beheading him: "Till he unseam'd him from the nave to the chaps, / And fix'd his head upon our battlements." Unseamed means cut open; from the nave to the chaps means from the navel to the chin; fixed his head means stuck his head; upon our battlements means on our fort's wall.

Later in the same scene, the Thane of Ross shows up and informs the king that the Thane of Cawdor is a traitor and was helping the rebels: "Assisted by that most disloyal traitor / The Thane of Cawdor...." The scene concludes with Duncan telling Ross to execute the traitor and give his title, Thane of Cawdor, to Macbeth.

This is how Macbeth realizes that the three witches' prophesy is true and later decides to kill the king. In Act I scene iii, The three witches meet Macbeth and Banquo on the battlefield and tell Macbeth that he will be Thane of Cawdor and then King of Scotland:

FIRST WITCH: All hail, Macbeth! hail to thee, Thane of Glamis!
SECOND WITCH: All hail, Macbeth! hail to thee, Thane of
Cawdor!
THIRD WITCH: All hail, Macbeth, that shalt be King hereafter!

Macbeth is currently the Thane of Glamis, but he does not believe he will become Thane of Cawdor and then king: "...to be King / Stands not within the prospect of belief, / No more than to be Cawdor" until he is informed that he has been promoted to Thane of Cawdor. In a way and without knowing it, by promoting Macbeth King Duncan has started the chain of events that will lead to his own death.

These quotes were taken from eNotes' text on Macbeth, which you can read here. The eNotes text has a modern English translation and annotations to make it easier to understand. 

How does Duncan reward Macbeth and Banquo?

Here's the part of the play you should be looking at. Duncan has just been told that the treacherous Thane of Cawdor repented on his death bed, and he is commenting regretfully on Cawdor's treachery as Macbeth himself comes in.

DUNCAN

There's no art
To find the mind's construction in the face:
He was a gentleman on whom I built
An absolute trust.

Enter MACBETH, BANQUO, ROSS, and ANGUS

O worthiest cousin!
The sin of my ingratitude even now
Was heavy on me: thou art so far before
That swiftest wing of recompense is slow
To overtake thee. Would thou hadst less deserved,
That the proportion both of thanks and payment
Might have been mine! only I have left to say,
More is thy due than more than all can pay.

It is some irony, isn't it, that Macbeth (who has already had "black and deep desires", presumably to kill the king) enters just as Duncan deplores the fact that you can't judge a book by its cover? Macbeth's face is already hiding a few treasonous secrets of its own!

Duncan in this speech expresses his gratitude to Macbeth and Banquo - and Macbeth, of course, we know is to be made Thane of Cawdor as a reward for his battle deeds. Shakespeare never specifies precisely what recompense Banquo receives, though we are told in this scene that he "no less deserves". Clearly, though, the role of Thane of Cawdor, is going to remain a problem one for Duncan!