Somewhere, not too far from the hero's hometown, there is something new and exotic to go see or do. The hero is fascinated with the idea of checking out this new thing, and he and his friends set out on a journey to do just that. Their journey drives the rest of the plot. This differs from characters on a quest, because there's no overarching need for them to take the trip, other than "let's go take the trip". They're tourists, not heroes out to destroy The One Ring. In addition, the actual object at the end of the journey is utterly unimportant, other than as a prod for the character to take the journey in the first place.
The name of the trope is from a 19th-century expression that meant "to take a trip to see or do something exotic." In The American Civil War era, it was sometimes used to refer to going to war. Sometimes used as a MacGuffin, making the character go in order to drive the plot, but not always, as it lacks the interchangeability of a true MacGuffin. Often, Its The Journey That Counts.
No relation at all to the Elephant in the Living Room, which everybody is pointedly trying to ignore. Also has nothing to do with going to see a man about a horse...
Samurai Champloo has the girl Fuu hire two Samurai, Mugen and Jin, to help her find the Sun Flower samurai. This Samurai in question does not show up until the last episode and the show regularly admits that he is nothing more than MacGuffin, with both Mugen and Jin regularly pestering Fuu about knowing nothing about the Sun Flower Samurai.
In Losin' It, each of the boys has a different "elephant" he wants to see during their trip to Tijuana, Mexico. One character wants to see a donkey show, another wants his car seats re-upholstered and the third just wants to get some cool fireworks.
The Straight Story is about a man who takes a cross-country trip on his riding mower, so he can reconcile with his dying brother.
Lost In America is about a couple who quit their jobs in order to "discover America." They do so by traveling around in a Winnebago on the back roads, looking for new things to see and do.
In Tangled, the entire reason Rapunzel leaves her tower is to go see the lights that appear on her birthday every year. They're more important than most examples though, as they are sent by her parents, the King and Queen of Corona, who want her to return more than anything.
In the Stephen King novella The Body (and its film adaptation, Stand by Me), the kids travel cross-country to see a dead body.
Barnaby Tucker, in Thornton Wilder's The Matchmaker (and its derivative, Hello, Dolly!!) has the fixed desire of seeing the stuffed whale in Barnum's museum when he goes to New York.
Interestingly, Maggie Antrobus wants to see a whale before she dies in the same author's The Skin Of Our Teeth. The Barnum Museum really was a tremendously popular place in the mid- to late-1800s, and the whale was a notable exhibit.
Mentioned by name in Terry Pratchett's Discworld novel Witches Abroad. It's even the last line, as the witches talk about how they're eager to get back home to Lancre: "But they took the long way round, and saw the elephant."
Another one that mentions the concept by name, "The Man Who Saw the Elephant" by Avram Davidson is about a farmer who goes in search of a rumoured travelling show featuring an elephant, because he's spent his entire life within a few miles of his farm and wants just once to go somewhere and do something out of the ordinary.
Starlight and Shadows trilogy. A young, talented and fun-loving drow wizard chanced upon a book about some forgotten surface people that mentioned an obscure tradition — rune magic. She thought it may be interesting to investigate the matter. Not that she expected to find a great power in it, just because it's unusual enough. The rest of three books cover some consequences of her curiosity.
The Epic of Gilgamesh: One of the literally oldest quests in the book is this trope — Gilgamesh decides to celebrate his new friendship with Enkidu by going to the Forest of Cedars and killing the guardian Humbaba for absolutely no logical reason, even in context. *
Gilgamesh later goes on a more goal-oriented heroic quest to find the secret of eternal life.
From Space Viking by H. Beam Piper, regarding Lucas Trask's first experience of being part of a Viking raid on a more-or-less helpless planet: "Well, you saw the elephant, Lucas. You don't seem to have liked it." Spoken by veteran Viking Captain Otto Harkaman.
There is a short story in the Honorverse called "Let's go to Prague" that starts out as being about a vacation. Of course it gets complicated from there.
Robert A Heinlein's "The Man Who Traveled in Elephants" is a literal inversion of sorts. A traveling salesman and his wife don't want to settle down in retirement. They need some nominal reason to justify roaming around the country, so imagine themselves to be selling elephants. (They figure they don't need to take an actual sample elephant to show, because, heck, everyone knows what an elephant looks like.)
Also literally inverted in Around the World in Eighty Days. During one leg of his journey, Phileas Fogg literally goes to see an elephant, but he's shown zero interest in sightseeing; he just wants to hurry up and get past a gap in the train routes.
On How I Met Your Mother, when Ted and Marshall were in college they used to make 21 hour road trips to visit a Chicago restaurant that served delicious but incredibly disgusting pizza. They do it again in their thirties when they find out the restaurant's about to close.
In The Drew Carey Show episode "New York and Queens", the gang travel to New York City to see the Yankees play the Indians. While driving cross country, they decide to take a side-trip into Hershey, Pennsylvania, just because they can smell the chocolate from the highway.
In Anyone Can Whistle, a large number of pilgrims appear once the "miracle" set up by the mayoress and her cronies gets going, and then at the end, they all (and a number of people from the town) leave when they hear that there's another miracle in the next town over.
On one episode of Rocko's Modern Life, Heffer and Rocko go on a road trip to see Phlegm Rock (a rock formation shaped like a runny nose, where the "mucus" is algae-ridden water spewing from a spring) before it closes and gets paved over.
In an episode of Doug, the Funnies are taking a road trip to see the Painted Gorge. They don't have any other plans, but end up doing stuff along the way.
In his personal memoirs, Civil War veteran Elisha Hunt Rhodes wrote that, after his experiences in the Civil War, he was no longer interested in aggression, belligerence, or anger, and would no longer own or use a firearm, because "I have already seen that elephant." He'd gone off to war thinking it would be a grand adventure, and it turned out to be just what war is: a horrible, terrifying, life-changing experience.
Many of gold seekers who went west during the Gold Rush hoping to hit pay dirt or the young trailhands going from Texas to the wild Kansas cowtowns described their journeys as "going to see the elephant."
Until 1896, there was an building literally shaped like an elephant on Coney Island which housed a brothel. The phrase "going to see the elephant" was (and still is in some pockets of New York City) used as a euphemism for visiting a prostitute.