Chapter 4 - "Horace Slughorn" - In which Harry and Dumbledore Apparate to Budleigh Babberton - discussing Voldemort, Inferi, and Dumbledore's injured hand along the way - and meet Slughorn, shortly after he has faked his own demise. Dumbledore asks Slughorn, to teach at Hogwarts, and Harry convinces him to while Dumbledore is out of the room. Dumbledore then takes Harry to the Burrow, where before entering he tells Harry about Slughorn, talks to him about Sirius and the prophecy, and tells Harry he will be taking private lessons from Dumbledore during the school year.
Dumbledore Calls on the Dursleys
A Walk Through Budleigh Babberton
Harry Meets Horace Slughorn
Arriving at [Slughorn's house], Harry and [Dumbledore] are met with a scene of "total devastation."
While [Dumbledore] is in the bathroom, Harry inadvertently convinces Horace Slughorn to return to Hogwarts as [Potions master]. When [Dumbledore] returns, Slughorn accepts the job, and [Dumbledore] and Harry head off into the night.
Dumbldore and Harry Outside of the Burrow
Before heading into the [Burrow], [Dumbledore] calls Harry into a [broom shed] to quickly discuss Sirius and the [prophecy] with him. He also tells Harry that they will be having private lessons together this year.
Chapter 5 - "An Excess of Phlegm" - In which Harry talks with Mrs. Weasley (and later Mr. Weasley) about the Ministry, we see Fleur and learn that she and Bill are engaged to his family's dismay, and Harry tells Ron and Hermione about Slughorn, his private lessons with Dumbledore, and the prophecy, while they tell him about Fleur and Tonks. O.W.L. results then arrive, with Harry earning seven O.W.L.s but realizing that Snape won't allow him to take N.E.W.T. potions, and therefore he won't be able to become an Auror.