Harry Potter and the Philosopher's/Sorcerer's Stone
The Mirror of Erised is used to hide the titular stone.
Harry's scar alerts him to Voldemort's presence, lets him see what Voldemort is seeing, and eventually is revealed to be from Voldemort's accidental Horcrux in Harry.
The flash of green light from the Killing Curse, Avada Kedavra.
The bezoar mentioned briefly in Harry's first Potions class is later used in Book 6 for Harry to circumvent an antidote challenge, and to save Ron after he accidentally drinks a bottle of poisoned mead.
Also used in Book 6 as a clue to the identity of the eponymous Half-Blood Prince.
The tongue-tying curse mentioned in Harry's first visit to Flourish and Blotts, which keeps Snape from telling the location of Grimmauld Place to the Death Eaters in Book 7.
Probably the one that holds the record for longest intro-to-relevance time is Dumbledore's deluminator, aka the "Put-Outer". Introduced in the first chapter of this book, it gets a throwaway cameo in Book 5, then becomes critical to the plot about halfway through Book 7.
The Snitch from Harry's first Quidditch match sat around for six-nearly-seven books before being useful for more than just being, well, an awesome catch. And the fact he caught it in his mouth is used to stretch it out a few extra pages.
In the scene where Harry goes to buy his wand, Olivander comments rather offhandedly that Harry's father, James, "favored a mahogany wand. Eleven inches. Pliable. A little more power and excellent for transfiguration." Well, good for him, because that surely must have come in handy later when he undertook becoming an Animagus, as is revealed in the third book.
In the same scene, he reveals that Harry's wand and Voldemort's share their core's source, which becomes important at the end of Goblet of Fire.
Hagrid mentions that his flying motorbike belongs to Sirius Black; Sirius isn't mentioned again until Book 3, and the motorbike itself doesn't come back until Book 7.
When Harry is attacked by Quirrel/Voldemort in the Forbidden Forest and is rescued by Firenze, the other centaurs object, citing "the stars." Even Firenze himself mentions to Harry that he hopes that he and his fellow centaurs have been misreading them. Considering the fact that centaurs are particularly gifted at astrology, they're probably seeing Harry getting killed by Voldemort in the Forbidden Forest at the end of Book 7, and don't want to interfere with destiny.
They also say 'Mars is bright tonight' several times. Mars is the god of war and bloodshed, and the planet symbolic of that. They're predicting the war that starts later on in the series.
Ron's spell to turn Scabbers yellow. Maybe Fred and George weren't wrong about it working on rats. They hadn't tried it on an Animagus.
Chekhov's Boomerang: The Dumbledore card from the Chocolate Frog, functioning in this book to inform the characters about Nicolas Flamel and in Deathly Hallowsabout Grindelwald.
Chekhov's Gift: The Invisibility Cloak, which Harry gets for Christmas, not only helps him throughout the series but turns out to be one of the titular Deathly Hallows in Book 7.
Rather hilariously, there was a book released a short time before the book that discussed Chekhov's Guns from the books. It concluded that, since so many people knew about it at this point (reasoning that Snape told Voldemort), the cloak had no more bullets left to fire. Boy was it wrong.
There is a scene in Harry Potter and the Order of the Phoenix where Moody mentions that his Invisibility Cloak is starting to lose its powers (hinting that Harry's cloak is special in some way, because it isn't losing its at all).
Hagrid mentioning that he was expelled from Hogwarts. We find out why in Book 2.
Hagrid explaining how secure Gringotts is, that dragons guard the high security vaults, and also how unwise it is to try double-crossing Goblins (or otherwise pissing them off).
Mrs. Figg is introduced as Harry's babysitter, and we hear very little of her afterwards. In Book 5, she's revealed to be a Squib, and is the lone witness in Harry's Ministry hearing.
She is also mentioned at the end of Book 4: When Dumbledore discovers Voldemorts return, he orders Sirius to contact old members of the order. He mentions Remus Lupin, Mundungus Fletcher, and Arabella Figg!
Grindelwald, mentioned in passing on Dumbledore's chocolate frog card, and not mentioned again until Book 7, where he's critical to the plot.
Petunia Dursley. While she's established as an important character, a reader may wonder why Dumbledore took Harry to live with his horrible, abusive relatives or why Harry in the later books even bothers to return to the Dursleys' home at the end of each school year. In Book 5, it's revealed that because she has Lily's blood in her, her presence protects Harry from Voldemort until he comes of age.
Sirius Black, mentioned in passing by Hagrid at the beginning of the book, and not mentioned again until he ends up being the perceived villain for most of Book 3.
Snape is introduced and is a major character, but it takes a few more books before we realise just how important he is.
Others include Griphook, Olivander and Scabbers.
First Girl Wins: Two cases. The first girl we see Ron meet to whom he isn't related is Hermione. Likewise when Harry first was on his way to Hogwarts, the very first girl we're shown with him is Ginny shyly trying to get a peep at him. Both of these come to fruition so late in the books that they definitely qualify.
The Wingardium Leviosa levitation charm is used to knock out the mountain troll.
Ron's skill at wizard chess also comes in handy.
Harry's ability to talk to snakes is assumed by Harry and the readers to be a manifestation of his wizard abilities, but we later learn it's Dark Magic accidentally granted to Harry by Voldemort.
The ability of Animagi to change themselves from human to animal form. Shown being done in this book only by McGonagall, but then turning out to be very important when done by other characters in the later books.
"Could Snape possibly know they've found out about the Philosopher's Stone? Harry didn't see how he could, yet he sometimes had the horrible feeling that Snape could read minds."
Snape is not actually that good at reading minds (Dumbledore and Voldemort are the masters of that). His main talent is that he is a master of preventing people who are skilled at reading minds, including Voldemort, from reading his mind, making him the perfect double agent.
This gets more attention in the film than anything, but the first part of Harry that Snape looks at is his eyes, not the scar. The seventh book reveals why.
Harry Potter and the Chamber of Secrets
Many things in this book (Tom Riddle, his diary, Ginny's feelings for Harry) come back and play an important role in book six.
Some examples: the Basilisk's fangs, which turns up again in book seven; the diary, which we later learn is a Horcrux; Ron's malfunctioning wand; the damaged Vanishing Cabinet that becomes important in book six.
Basilisk Fangs. Turns out Basilisk venom is one of a very few ways to destroy a horcrux. When Harry uses the sword of Gryffindor to kill the basilisk, he imbues it with said venom. Dumbledore later uses the sword to destroy the ring horcrux, Ron then uses it to destroy the locket horcrux, and Neville uses it to destroy Nagini, the last remaining horcrux at the time. Hermione and Ron later return to the Chamber and Hermione uses a fang to destroy the Hufflepuff Cup horcrux.
Chekhov's Skill: "[Phoenixes] can carry enormous weight, their tears have healing powers..."
They also regularly die and then return to life, which allows Fawkes to intercept Voldemort's Avada in book 5, Taking the Bullet for Dumbledore.
Mundungus Fletcher is mentioned in passing by Arthur for having tried to hex him during a raid. Mundungus becomes important in books five through seven.
Hagrid tells the trio that the DADA position is rumored to be jinxed in this book. Sure enough, no DADA teacher in the subsequent books lasts longer than a year, either because Voldemort cursed the job when he was denied it.
At Borgin and Burkes, Malfoy notices the Hand of Glory, a magical artifact that gives light only to the holder. He eventually uses it in Book 6 for his plan to let Death Eaters into the school.
The same scene also mentions the cursed necklace and, though it's presented as no more than a regular cupboard in this scene, the Vanishing Cabinet, both of which also become major plot-points in The Half-Blood Prince.
Chekhov's Boomerang: The flying car is used to take Harry from the Dursley's house. Then again to fly to Hogwarts and then to escape from the Forbidden Forest later.
Harry's letter from the Improper Use of Magic Office was sent by Mafalda Hopkirk, who makes a reappearance in book 7, when Hermione uses her hair in Polyjuice Potion to infiltrate the Ministry.
Polyjuice Potion itself. It ends up becoming a method by which a Death Eater is able to infiltrate Hogwarts in Book 4. In Book 7, it's part of the plan to safely relocate Harry from the Dursleys', as well as how the Trio infiltrate the Ministry of Magic.
The Ghoul, who Ron mentions lives in the attic, shows up in book 7 when he's transfigured into Ron so Ron can go with Harry and the family avoids suspicion by the Ministry
The plaque Ron barfs slugs all over during his detention turns out to be an important clue later in this same book.
Chekhov's Classroom: The Herbology lesson on mandrakes establishes their curative effects before anyone at Hogwarts actually needs them. The fact that fantastic creatures' death-inducing powers can be rendered non-lethal under the right circumstance (mandrakes too young) is also slipped into the lesson: a fact that will save a lot of lives that year.
Chekhov's Skill: Ron, while convincing Harry to take the Ford Anglia, mentions apparation, which he and Hermione formally learn in book 6 and becomes very important in book 7.
Also, Harry learns his "signature" spell (from Snape, of all people!) at the Dueling Club that occurs in this book, which ends up being a gift that keeps on giving. The idea of Snape as a Stealth Mentor (sometimes unintentionally) comes up big time in Book 6. Also, the idea of an informal group learning dueling skills ends up becoming a big deal in Book 5 when Harry becomes the leader of one. And guess what the first spell is that he teaches his group? Also, the group is reborn in Book 7 and really gets a lot of mileage out of all the stuff Harry taught them...
Harry Potter and the Prisoner of Azkaban
The newspaper photo of the Weasley family's vacation is used by Sirius Black to realize Pettigrew is alive.
Also Scabbers, AKA Peter Pettigrew, and to a much lesser extent Crookshanks.
Professor Trelawney, at the end of the book, Dumbledore casually mentions that he should give her a raise, since the prediction she made about Wormtail turned out to be true, and that it was the second accurate prediction she had ever made. At the end of OotP, it's revealed what the first one was.
Snape seemed to be the only teacher zealously trying to find Sirius. He's even willing enough to kill him and eager enough to watch the dementors suck out his soul. We find out that Sirius was a bully in the fifth book, and that Snape, like most people, thought that Sirius had led Voldemort to James and Lily. The seventh book reveals Snape's feelings for Lily.
Also, quite a few of the predictions done in the Divination Classroom turn out to be truer than one would've thought!
Ron: So...you're gonna suffer...but you're gonna be happy about it...
Just a few sentences later we also know that Harry will work for the ministry (As an auror, no less), get unexpected fortunes (from winning the Triwizard tournament the next year) and die. Even if it happens a lot later (and under different circumstances) than expected.
Trelawney also predicts that Hermione drops out of Divination class by Easter. Even if she probably did not know it herself at the time. * She predicted that "One of our number will leave us forever". She misinterpreted it as a Deadly Euphemism.
Trelawny continues to predict a grim in Harry's life. A Grim resembles a large black dog according to this book...and boy howdy does a large black dog (Sirius Black's animagus form) play a part in Harry's life by the end of this book and through book 5.
Furthermore, she claims the grim is an omen of death. It is. Just not for Harry.
Trelawny also tells Parvati Patil to beware a red-haired man. Ron goes out with her sister the next book...
The spy who tipped Dumbledore off that Voldemort was going after the Potters was Snape.
The sneakoscope that Ron thought to be broken because it kept spinning when Scabbers was near it.
A more specific example related to Bertha Jorkins, her having been traveling in Albania when she was caught, her Memory Charm broken, and she was forced to divulge the truth about Barty Crouch, Jr. Why? Albania is where Helena Ravenclaw hid her mother's diadem, and where Voldemort later found it so as to take it to the Room of Requirement. And it's highly likely this is where Quirrell was traveling when he encountered Voldemort's spirit and became corrupted/possessed; he was supposed to have been in the Black Forest, which shows he was already moving around Europe.
Chekhov's Gunsnake: Nagini is introduced in the seemingly unrelated first chapter and is not featured significantly until the next book, when Voldemort orders her to attack Arthur Weasley and Harry is able to see the scene not only through Voldemort's eyes, but through Nagini's, possibly because they're both Voldemort's Horcruxes. To return to the current book, though we don't know it until late in Half-Blood Prince, this apparent filler chapter is in fact showing the moment when Voldemort turned Nagini into a Horcrux.
Dumbledore's brief look of triumph when he is told that Voldemort used Harry's blood to resurrect himself. Dumbledore just realized that Voldemort had just made it possible for Harry to survive the Killing Curse again.
The flash of green light Harry kept remembering in the first book. Revealed in this book to be the killing curse.
Neville reacting badly to the Cruciatus Curse. Later in that book, we find out why...
Accio spell. Taught in Flitwick's class earlier in the book. Harry struggles with it and then finds out it's the only way to get his broom to him to survive the first task. He eventually gets it right and uses it successfully not only in the tournament but to call the Tri-Wizard cup to him in the graveyard allowing him to take Cedric's body and himself back to Hogwarts.
During the Yule Ball, Dumbledore makes mention of having a Potty Emergency and discovering a room with "a really rather magnificent collection of chamberpots. When I went back to investigate more closely, I discovered that the room had vanished". Perhaps, Dumbledore theorizes, the room only appears when somebody really, really needs it...
After Hagrid is embarrassed by Rita Skeeter, and wishes to resign because of the public disgrace, Dumbledore tells him that his brother Aberforth had been convicted for practicing inappropriate charms on a goat. This is the first mention of Aberforth, but he becomes vitally important in Book 7.
Another one from Trelawney: She tells Harry "That thing you have been dreading will come to pass". This can actually mean two instances: Voldemort returns in the end of the fourth book, and also, in the next book Sirius dies
A throwaway line from Amos Diggory at the beginning, when he meets Harry, Hermione and the Weasleys to go to the Quidditch World Cup: "The Lovegoods have been there for a week already." We think these are just random people they know until meeting Luna in the next book and her father in Book 7.
Harry Potter and the Order of the Phoenix
The once-mentioned locket that nobody could open when Number 12 Grimmauld Place was being cleaned out, which became important in book seven.
Harry begins having dreams early on in the story about locked doors and winding passages.
A portrait at St Mungo's insists Ron must have spattergroit; Ron retorts he's just freckled. In Deathly Hallows, Ron fakes being near death with spattergroit to follow Harry on the quest for Horcruxes and buy time to avoid Death Eaters coming after his family for helping Harry.
The thestrals, and Harry being able to see them.
Hermione makes a list containing the names of all the people agreeing to be in the DA. It's mentioned it that signing it feels unusually binding. Hermione cursed the list so that anyone who revealed the DA to the Ministry would break out in a rash. Not just any rash, either. It (permanently, per Word of God) writes 'Sneak' across their forehead.
In Chamber of Secrets, a passing mention was made to Peeves breaking a cabinet, preventing Harry from getting into trouble. It's mentioned here again, when the Weasley twins push one of Umbridge's hench-students in, and he gets trapped there for months because it's magical. In the next book, it's revealed to be linked with another cabinet outside the school, and Draco fixes it to let the Death Eaters in.
"Snape's Worst Memory". It involves Snape being publicly humiliated by Siruis, James and Peter, and provides an interesting Alternative Character Interpretation to James Potter (Jerk Jock, specifically). It's pretty bad on the surface... but in Book 7, we learn why it was his worst memory: it was the day he lost Lily forever (due to his own behavior, not really anything James did).
Chekhov's Gift: Near the middle of the book, Sirius gives Harry a magic mirror which is part of a pair that can enable the two holders to communicate magically. Sirius has the other one. Made horrifying when you realize that the driving point of the climax of the book is that Harry is trying to communicate with Sirius, but can't find a means to; only after Sirius has been murdered does he find the mirror in his school trunk. It's thrown away and broken, but a fragment of it becomes important in Deathly Hallows
Chekhov's Gunman: Mrs Figg, introduced in Book 1 as Harry's baby-sitter, and then not mentioned again until briefly in Goblet and this book, which reveals her to be a squib and is the witness at Harry's trial who gets him acquitted.
The barman of the Hog's Head Tavern, identified two books later as Albus Dumbledore's younger brother Aberforth, who had been mentioned in passing by Dumbledore earlier. It was even noted that he "looked vaguely familiar".
Regulus Black, Sirius's brother who is mentioned in passing and the becomes important to the plot in Book 7.
Kreacher, a seemingly minor character who, in true JK Rowling style, ends up being important to both this book's plot and that of Deathly Hallows. When screenwriter Michael Goldenberg suggested writing him out of the movie adaptation, Rowling admitted, "If you do that, adapting Seven's going to be really difficult..."
Professor Trelawney. In Book 3 Dumbledore casually mentions that he should give her a raise, since the prediction she made about Wormtail turned out to be true, and that it was the second accurate prediction she had ever made. At the end, it's revealed what the first one was.
This is also the book where we start to see Neville's emerging combat skill, along with his determination, both of which become of great importance in Deathly Hallows.
This book also contains an example of what happens when you overuse Chekhov's Gunmen. Harry refers in passing to a Mark Evans while speaking to Dudley. This is a throw-away character who is never mentioned again, but many assumed he would have some importance in the future because he shares a same last name with Lily Potter (née Evans) and Petunia Dursley (née Evans). JK Rowling publicly apologised for it.
Chekhov's Skill: Occlumency. Harry never masters it, but Snape's talent in it is vital to his role as a spy.
The Quibbler, and the fact that Luna Lovegood's father is the editor. We meet him in Book 7.
Harry Potter and the Half-Blood Prince
When Harry hides his potions book in the Room of Requirement he marked the spot with a bust of a warlock wearing a wig and an old, battered tiara. Said tiara becomes immensely important in Book 7 when it's revealed to be the Lost Diadem of Ravenclaw, one of Voldemort's Horcruxes.
To add to that, the Room of Requirement itself. Draco was using it, specifically the Room of Hidden Things, to try and find a way to kill Dumbledore. The room is revealed in Book 7 to be the location of one of the Horcruxes, and Draco comes close to foiling Harry there because of how much experience he has with that room - as he says, "I virtually lived in the Room of Hidden Things last year." He knows how to get in.
Tonks' Patronus changing its form. The fact that it's supposed to represent Lupin shows that the Patronus often represents someone the caster loves... much like Snape's.
During this years' only Hogsmeade trip, the trio encounter Mundugus again, selling things he stole from Grimmauld Place. The Hog's Head barman (Aberforth) was talking to him as they approached, and most likely acquired Sirius' mirror at that time, which becomes quite significant in the next book.
Harry Potter and the Deathly Hallows
Harry's snitch, Dumbledore's Deluminator (aka the Put-Outer), Godric Gryffndor's sword, dragons at Gringotts, the poem warning thieves at Gringott's entrance, the eye Harry sees in Sirius's mirror, the tiara Harry put on a bust to mark a spot in the Room of Requirement, the locket in the House of Black, Wormtail's hand, Voldemort's claim about Wormtail's loyalty, Wormtail's debt to Harry, Harry's invisibility cloak, and the Basilisk's fangs.
Not to mention the protection charm cast by Lily which protect Harry from Voldemort. Harry's Heroic Sacrifice fuels the same charm, protecting all the students and staff of Hogwarts.
Chekhov's Gag: Movie-only. In the movies, Seamus had a Running Gag where he'd blow things up accidentally (usually in the background). Come the Battle of Hogwarts and McGonagall knows exactly who to turn to to blow up a bridge.
Trelawney's continued obsession with Harry's death finally comes to fruition in this book, though it's the one year where they don't interact until the very end and it doesn't take anyway.