Best
tl;dr: 5 times
Harry mentions in multiple locations that this is "the best day", I will outline these examples following. Bear in mind that "the best ever" is a common term which young people use when feeling elated (same goes for the converse "worst ever"); we like to exaggerate.
Harry Potter and The Philosopher's Stone
The Dursleys and Harry visit the Zoo in Surrey to commemorate Dudley's 11th birthday. Harry seemed to enjoy the visit to the Zoo and felt that it was one of the best days he'd had in a while:
[the lemon ice pop] wasn’t bad, either,
Harry thought, licking it as they watched a gorilla scratching its
head who looked remarkably like Dudley, except that it wasn’t
blond.
Harry had the best morning he’d had in a long time. He was
careful to walk a little way apart from the Dursleys so that Dudley
and Piers, who were starting to get bored with the animals by
lunchtime, wouldn’t fall back on their favorite hobby of hitting
him...
-Harry Potter and The Philosopher's Stone, Chapter Two (The Vanishing Glass).
Harry Potter and The Philosopher's Stone
Harry finds out that he is a wizard on his 11th birthday. He feels that he's had the best birthday ever:
“You all right, Harry? Yer very quiet,” said Hagrid.
Harry wasn’t sure he could explain. He’d just had the best birthday
of his life — and yet — he chewed his hamburger, trying to
find the words. ...
-Harry Potter and The Philosopher's Stone, Chapter Five (Diagon Alley).
Harry Potter and The Philosopher's Stone
Harry's first Hogwarts Christmas, and it turned out to be his best Christmas so far:
It had been Harry’s best Christmas day ever. ...
-Harry Potter and The Philosopher's Stone, Chapter Twelve (the Mirror of Erised).
Harry Potter and The Philosopher's Stone
Harry's first End of Year Feast. After defeating Quirrelmort, Harry attends the End of Year Feast in which Prof. Dumbledore announces Gryffindor as the Cup winners:
It seemed as though life would be back to normal next year,
or as normal as it ever was at Hogwarts.
It was the best evening of Harry’s life, better than winning at
Quidditch, or Christmas, or knocking out mountain trolls . . . he
would never, ever forget tonight. ...
-Harry Potter and The Philosopher's Stone, Chapter Seventeen (The Man with Two Faces).
Harry Potter and The Prizoner of Azkaban
Harry and his team practice for their match against Ravenclaw that month and during one of the practices, Harry says that it was the best Quidditch practice ever:
It was the best practice ever; the team, inspired by the presence
of the Firebolt in their midst, performed their best moves faultlessly ...
-Harry Potter and The Prisoner of Azkaban, Chapter Thirteen (Gryffindor versus Ravenclaw).
Worst
tl;dr: 4 times
Harry Potter and The Philosopher's Stone
It was lucky that Harry had tea with Hagrid to look forward to,
because the Potions lesson turned out to be the worst thing that
had happened to him so far.
-Harry Potter and the Philosopher's Stone, Chapter Eight (The Potions Master).
Harry Potter and The Chamber of Secrets
Funnily enough, the first chapter is called "The Worst Birthday"; but there's no mention by Harry about the worst of anything.
Harry Potter and The Chamber of Secrets
When Ginny got abducted by the monster in the Chamber of Secrets:
“The Heir of Slytherin,” said Professor McGonagall, who was very white, “left another message. Right underneath the first one.
‘Her skeleton will lie in the Chamber forever.’ ”
Professor Flitwick burst into tears.
“Who is it?” said Madam Hooch, who had sunk, weak-kneed,
into a chair. “Which student?”
“Ginny Weasley,” said Professor McGonagall.
Harry felt Ron slide silently down onto the wardrobe floor beside
him.
“We shall have to send all the students home tomorrow,” said Professor McGonagall. “This is the end of Hogwarts. Dumbledore always said . . .”
...
It was probably the worst day of Harry’s entire life. He, Ron, Fred, and George sat together in a corner of the Gryffindor common room, unable to say anything to each other. Percy wasn’t there. He had gone to send an owl to Mr. and Mrs. Weasley, then shut himself up in his dormitory.
-Harry Potter and the Chamber of Secrets, Chapter Sixteen (The Chamber of Secrets).
Harry Potter and The Goblet of Fire
When Ron stopped talking to him because everyone thought he had put his name IN THE GOBLET OF FYAH!
The next few days were some of Harry’s worst at Hogwarts. The closest he had ever come to feeling like this had been during those months, in his second year, when a large part of the school had suspected him of attacking his fellow students. But Ron had been on his side then. He thought he could have coped with the rest of the school’s behavior if he could just have had Ron back as a friend, but he wasn’t going to try and persuade Ron to talk to him if Ron didn’t
want to. Nevertheless, it was lonely with dislike pouring in on him from all sides.
-Harry Potter and the Goblet of Fire, Chapter Sixteen (The Chamber of Secrets).