The Hanged Man
In the first chapter, Frank Bryce, the unfortunate gardener for the Riddle house, is arrested for the murder of the Riddle family in Little Hangleton, England. Frank is soon released because there's zero physical evidence to show how the Riddles were murdered. Although Frank claims to have seen a teenage boy murder the Riddles, no one believes him. In fact, the villagers are absolutely against Frank. Even though no one officially accuses him of anything, everyone seems to agree with village resident Dot when she comments, "So far as I'm concerned, he killed them, and I don't care what the police say" (1.23). So Frank has been convicted of murder according to public opinion, even if there's no real evidence against him.
This is why we find the name of the village pub, the Hanged Man, to be significant. A hanged man is (generally) a man who has been convicted and executed of a crime. It's in the Hanged Man pub where all of these unfounded rumors against Frank Bryce start circulating, so the name of this pub foreshadows what Frank Bryce is. Oh, he hasn't been executed (at least, not until later on in Chapter 1), but he has definitely been convicted, if only unofficially in the minds of all the villagers.
Harry's Curse Scar
Harry's lightning bolt-shaped scar on his forehead becomes a major plot point in Harry Potter and the Goblet of Fire. The scar is the only mark on Harry's body from when Voldemort tried to kill him with an Avada Kedavra curse when he was a baby. It's also always been a way for strangers to identify Harry, whether he wants to be recognized or not.
But now, it's more than just proof of what happened to Harry thirteen years before – it's a tangible symbol of Harry's magical connection to Voldemort. The horrible pain that Harry feels when Voldemort casts the Cruciatus Curse is concentrated right on this scar. And his dreams of Voldemort's activities also cause his scar to ache. Professor Dumbledore puts forward a theory, and we trust his word:
It is my belief that your scar hurts both when Lord Voldemort is near you, and when he is feeling a particularly strong surge of hatred [...] Because you and he are connected by the curse that failed [...] That is no ordinary scar. (30.164-6)
We know that Ron is afraid of spiders – we find out when Harry and Ron head out to the Forbidden Forest in Harry Potter and the Chamber of Secrets and meet a giant spider (hey, anyone would be afraid of that). When Professor Moody brings out his jar of three spiders to demonstrate the Unforgivable Curses, Ron is frightened even before Professor Moody demonstrates the curses. So, in this scene, the spiders are a symbol of fear; they underline how terrifying these curses are by their very nature.
The Goblet of Fire and the Triwizard Cup
"Goblet of Fire" just sounds cool; who wouldn't want to see a goblet of fire? For more on the origin of this name, check out "What's Up With the Title?" The Triwizard Cup is the goal for which Harry is competing. It symbolizes his reward for all the struggle and hardship he has endured as the youngest unintended champion in the Tournament.
What's interesting about both the Goblet of Fire and the Triwizard Cup is that both of them are smokescreens. They seem like they should be important in and of themselves – their names are capitalized, after all. But their primary significance is what they lead to: the Goblet of Fire is enchanted into choosing Harry as a Triwizard Tournament competitor, and the Triwizard Cup becomes a Portkey to bring Harry and Cedric to the Little Hangleton graveyard where Voldemort is waiting. These two objects are red herrings, things that look like clues but really aren't. You can easily forget about the Goblet of Fire by the end of Chapter 16. It's a means to an end, as is the Triwizard Cup.
We know from Book 1 that Harry and Voldemort's wands share the same core – phoenix feather. (In Goblet of Fire, we find out that the phoenix in question is Dumbledore's dear Fawkes.) The shared wand core has huge plot significance in Goblet of Fire, since that's what triggers the Priori Incantatem effect that allows Harry to get away from Voldemort. It's also symbolically significant in the same way Harry's curse scar is: Voldemort and Harry are linked by their magic. That's how we know that Voldemort is going to keep coming up as part of Harry's destiny, no matter how many times Harry escapes or delays their inevitable showdown. What's more, phoenixes are known for coming back to life even when everything seems lost. This is something that both Harry and Voldemort (regrettably) have proved able to do.
In general, wand cores say something about the personality of the wizard or witch who carries it. Fleur Delacour's wand uses hair from her veela grandmother's head. Mr. Ollivander comments that veela hair "makes for rather temperamental wands" (18.172). And Fleur is pretty darn temperamental herself, with her 180-degree turnaround on what she thinks of Harry. Cedric's wand comes from the tail hair of "a particularly fine male unicorn" (18.176), and Cedric is a particularly fine young man. All the associations with unicorns are good: they're brave and pure of spirit. Cedric seems to share those qualities. Last but not least, Viktor Krum's wand contains dragon heartstring and is "quite rigid" (18.183). He is also stiff, awkward, and somewhat dangerous, but he's brave and strong, a lot like a dragon.
The Single Tissue from the Dursleys
By Goblet of Fire, we don't really know why the Dursleys insist on giving Harry Christmas presents – they might as well save themselves the effort. In Book 4, they hit "an all-time low" by giving Harry "a single tissue" (23.49). The simple fact that the Dursleys bother to send Harry a single tissue, rather than just ignoring him entirely over Christmas, suggests that their gift-giving is meaningful. We just don't know what it symbolizes yet. We'll just have to keep reading the series to find out!