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
[Dumbledore] sends [Harry]'s things to the [Burrow] and they prepare to depart from [Privet Drive].
A Walk Through Budleigh Babberton
[Harry] and [Dumbledore] leave [Privet Drive] together, and after walking a block or so away, [apparate] to [Budleigh Babberton].
[Harry] and [Dumbledore] walk through [Budleigh Babberton] toward [Slughorn's house], discussing the [Ministry of Magic] and the [Daily Prophet] along the way.
Harry Meets Horace Slughorn
Arriving at [Slughorn's house], [Harry] and [Dumbledore] are met with a scene of "total devastation."
[Dumbledore] discovers [Horace Slughorn] hiding in his house, and together the two clean the place up, then sit discussing [Slughorn]'s life.
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
As they walk away from [Budleigh Babberton], [Dumbledore] tells [Harry] more about [Slughorn]. They then [Apparate] to 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.