Neville's Remembrall turns red, indicating he's forgotten something; come that night he's stuck outside the Gryffindor common room having forgotten the password, and therefore has to go with the Trio to the "Midnight Duel".
After it's revealed how the new students are sorted into houses, Ron mentions that Fred "was going on about wrestling a troll" when describing how the sorting worked. Not yet, Ron, but in a few more chapters...
When Hagrid explains the professors who arranged for the traps to protect the Mirror of Erised, and likewise the Philosopher's Stone, Quirrell is among those listed. Later, when they are proceeding through the traps to get to the Philosopher's Stone to prevent who they think is Snape from getting it, they enter Quirrell's trap room and discover a knocked out Troll, which gives away an early clue that Quirrell was the one who released a troll into the castle on Halloween, and that it is in fact Quirrell, not Snape, who was searching for the Stone.
Foreshadowing for future books:
Early on, Harry has a pleasant conversation with a boa constrictor. The next book reveals that this is a very rare (and somewhat notorious) ability, later explained by the fact that he's a Soul Jar for perhaps the most dangerous Parseltongue who ever lived.
One of the first things Harry learns about the wizarding world is that it's dangerous to mess with goblins and flat-out mad to try and rob Gringotts, but Voldemort manages to attempt this a few chapters later, offscreen. Then Book 7 rolls around...
The high-security vaults are guarded by dragons.
Early on, Hagrid off-handedly mentions that the current Minister of Magic, Cornelius Fudge—who won't appear in person until the next book and won't become particularly important until even after that—has a bit of a reputation as a bungler and is always writing to Dumbledore to ask for advice. Indeed, when we get to know him better, Fudge does turn out to be quite incompetent and would rather sweep a problem under the rug than actually deal with it properly. All of that comes back to bite him and the whole Wizarding World when it eventually leads to Voldemort taking over the Ministry.
Vernon and Petunia glare at Harry when he says he leaves from "platform nine and three-quarters" but only Vernon calls the idea of the weird platform "rubbish". We learn in Book 7 that Petunia has actually stood on the platform. She was probably quiet because she was reminded of it for the first time in years.
Even subtler, when Hagrid shows up at the island where the Dursleys are holed up, Vernon calls Dumbledore an "old fool". How would he have any idea whether Hogwarts' Headmaster is old or not? When he and his wife discussed Harry's letter, Petunia must have told him that Dumbledore worked there when Lily was at school.
When Harry goes to buy his wand at Ollivander's, Ollivander tells him about his parent's wands, mentioning that his father's wand was good for Transfiguration. In Book 3, we find out that James managed to secretly become an Animagus, which would require very good skills in transfiguration.
Harry claiming that Voldemort will turn Hogwarts into a school for the Dark Arts. Come Book 7, he does just that.
On the train to Hogwarts, Ron says "Scabbers might have died, and you wouldn't know the difference."
Midway through the book, Harry has a horrible feeling that Snape can read minds. In Book 5, Snape is revealed to be a skilled Legilimens, which happens to be the sort of mind-reading Harry mentioned all the way back in this book. However, when he explains this to Harry, he calls it "the ability to extract feelings and memories from another person's mind" and takes exception to Harry's characterization of this ability as "mind-reading". In the film, this is particularly notable when Snape looks into Harry’s eyes towards the end, and grows visibly alarmed before hurriedly running off. It’s now clear he used Legilimency on Harry to figure out what was going on with the Stone.
Quirrell mentions that the turban was given to him by an African prince for ridding him of a rather troublesome zombie — in Book 6 it's revealed that zombies are actually called "Inferi". And after reading Book 6, the reader can tell just how much Quirrell is lying.
Harry dreams about wearing Quirrell's turban, and it talks to him. He finds out later that the turban is concealing the face of Voldemort, who has possessed Quirrell. It's also an early indication that Harry has a connection with Voldemort's mind.
Bane gets angry about his fellow centaur Firenze saving Harry from Voldemort in the Forbidden Forest because it supposedly interferes "with what the planets say is going to happen." From this, Harry morbidly assumes his own doom is inevitable, saying the planets "must show that Voldemort's coming back... Bane thinks Firenze should have let Voldemort kill me... I suppose that's written in the stars as well." While these grave events are prevented from happening in Book 1, the planets are ultimately revealed to have been quite right: Voldemort does come back (in Book 4) and does kill Harry (in Book 7) — in the Forbidden Forest, no less.
Harry tried asking Dumbledore what the latter sees in the Mirror of Erised. When returning to bed, Harry realized that Dumbledore wasn't being truthful with his answer, and then telling himself that it was something personal for Dumbledore. Come Book 7, his suspicions are proven correct.
Hagrid mentions he bought Norbert from the Hog’s Head, the sketchy bar in town. It’s mentioned several more times until Harry and co. go there for a clandestine meeting in book five and get busted because it’s owned by Dumbledore’s brother. They don’t find out who he is until very late in book seven.
Voldemort correctly surmising that Harry lied to Quirrell about having the Stone during the climax at the Mirror of Erised. It seems like he's just being smart or that Harry didn't lie too well. Voldemort's actually employing Legilimency, though it won't be identified as such until Book 5.
Voldemort notes that he was after Harry specifically that night, and only killed Lily for defending him. When Harry asks Dumbledore why Voldemort wanted him dead, Dumbledore replies that Harry will be ready for the answer when he's older.
Right there on the Chocolate Frog Card: Dumbledore is particularly famous for his defeat of the Dark Wizard Grindelwald in 1945, which is when he scored the Elder Wand.
Related to this, Dumbledore praises Neville for having the courage to stand up to his friends, which takes more courage than standing up to one's enemies. In book 7 we find out exactly how Dumbledore learned that lesson.
Hagrid says that he doesn't think Voldemort is really dead because he doesn't think there was enough human left in him to die. Voldemort's body died with only part of his soul after splitting it, so he wasn't completely human, which is why he was able to come back.
Dumbledore says that one shouldn't spend so much time dreaming that you forget to live. His overly-idealistic dreams eventually led to his sister's death, which he's never really gotten over.
One of the lingering questions remaining at the end of the first book is why Voldemort ever wanted to kill Harry in the first place. This becomes a major plot point in the fifth book.
Hagrid borrows that flying motorbike from someone called Sirius Black. We'll meet him soon enough.
When Harry learns someone is killing and drinking unicorn blood for regaining some strength, which also gives the drinker a cursed life, Harry immediately says 'Living like that, death is better, isn't it?' Not only is the drinker revealed to be Voldemort himself, but in Book 7, according to Word of God, Voldemort's final fate is being stuck in Limbo, neither alive or dead, unable to pass on.
Fred and George sing the Hogwarts song as a funeral dirge. Fred dies in book 7.
Ron noticeably lacks enthusiasm when Fred and George mention wanting to see Lee Jordan's giant tarantula. The next book reveals Ron has arachnophobia as a result of some revenge-focused accidental magic from Fred.
During the sorting ceremony, the Sorting Hat tries to convince Harry that he would make a good fit for Slytherin House, but yields when Harry insists on being sorted into Gryffindor. The Sorting Hat was channeling either Harry's understated Slytherin traits or the horcrux of Voldemort — himself a Slytherin — embedded inside Harry.
Harry's first conversation with Draco Malfoy in Diagon Alley gives us the first hints of the Wizarding World's Fantastic Racism when Malfoy asks if Harry's parents were "our kind" (wizards) and expresses his belief that Hogwards shouldn't let the "other sort" (Muggleborns) attend. The concept of blood purity and anti-Muggle/Muggleborn prejudice would get explored in detail starting in the next book, and would become a large part of the series as one of the main tenets of Voldemort and his followers.
