These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Big Lipped Alligator Moment: The school song. No, not the Sorting Hat's song — the one that goes "Hogwarts, Hogwarts, Hoggy Warty Hogwarts." It is sung at the end of the beginning-of-term feast, but only in this book. It shows up very briefly in the fourth movie, with a more substantial inclusion in thisDeleted Scene from the same, but it's never mentioned again in the books. According to Word Of God, Dumbledore decides whether or not to have the school sing it based on whether he's in the mood.
Fridge Horror: When the first movie came out, there was a slew of merchandising, including a plushie of Ron's pet rat Scabbers. Who, two books/movies later, is revealed a disguised Death Eater who was responsible for getting one of his best friends (Harry's dad) killed by Voldemort and framing another (Sirius Black) for the crime. I'm sure every child who bought one could empathize with Ron.
One moment that could be this or (even more) Hilarious in Hindsight is the passage in the Christmas chapter describing the Weasley Twins enchanting snowballs to bounce off the back of Professor Quirell's turban after you've read the end of the book and discovered what he's keeping under there.
Another passage involves Hagrid forbidding Harry to buy a solid-gold cauldron. Imagine if Harry did buy that kind of cauldron, y'know, for Potions class...
It Gets Better: Notably, It takes about a third of the novel before Harry gets to Hogwarts. And quite a few children readers were turned off by the first few chapters for being too slow-paced; not to mention most of Rowling's dry satire on middle-class insecurities flying straight over their heads.
Ship-to-Ship Combat: Probably few people realized just how big and vicious of a battle this book would begin among its fans, most prominently among Harry/Hermione and Ron/Hermione shippers.
Squick: Dumbledore's experience with the Every Flavor Beans. He lost his liking to them a long time ago because of a vomit-flavored one, and when we see him actually try another, it was earwax-flavored.
What an Idiot: Hagrid takes Harry, Ron (only in the film), Neville, Hermoine and Malfoy into the Forbidden Forrest to look for a Unicorn Murderer for detention. Harmful to Minors does not begin to describe Hagrid.
Vernon insults magic and threatens Harry in front of Hagrid. I repeat, he insults magic and threatens Harry in front of a half-giant with super strength, who happens to be from the wizarding world. It's a good thing Hagrid isn't malicious, or Vernon would've probably ended up with a broken neck, or worse.