This marks the second time production filmed on location for a multi-episode arc, this time on the Hawaiian island of Kauai. Jon Stone said, "Going to Hawaii was very good for us because it is a multi-ethnic society and one that provides an ocean-oriented milieu that we have never had on the series." Three of the children who participated in the Hawaii shows are direct descendants of King Kaumualii, and two are descendants of King Kamehameha.
After a preamble of episodes in which the cast prepare for the trip, the story follows Big Bird, Oscar the Grouch, Maria, Mr. Hooper, Olivia, and Bob spending a week visiting with Buffy and her husband Sheldon on the Indian reservation where they live. Many of the themes explored in the story were centerd around Native American culture; or, as Buffy often referred to her people, "Indians" (including the song "I'm an Indian Wherever I Go"). A recurring subplot involves Mr. Snuffleupagus traveling to the island — also established as the indigenous home to snuffleupaguses — where he and Big Bird search tirelessly for Mount Snuffleupagus, known to the locals as Mount Ihu Papa'a Lo'ihi Nui.
The themed collection of episodes also prompted the first major departure in music during the show's opening. Beginning with the third day on the trip, the standard "Sesame Street Theme" was replaced with a rearranged Polynesian version comprised of English and Hawaiian lyrics. Sung by the cast, the music played over a montage of aerial footage taken above the Hawaiian islands.
Maria receives a letter from Buffy with photos of her home in Hawaii and an invitation to come visit. Several of her neighbors are interested, so they decide to hold a street fair to raise money for the trip: Luis and David run a taco stand, Maria sells old appliances, Mr. Hooper sells toys and books, Susan and Gordon sell balloons, and Olivia offers her services as a photographer. Oscar is also convinced to come along when he learns about the volcanic rock and mud that make up the islands of Hawaii.
When he learns that Mr. Snuffleupagus has never been to his place of origin, Big Bird invites him on the upcoming Hawaii trip. Snuffy explains that he can't afford the airplane ticket, but, based on his mommy's descriptions, he imagines what it's like all the time (with on-location footage shown over his narration). To cheer him up, Big Bird decorates the tree in front of his nest with coconuts in an attempt to emulate a Hawaiian landscape, and makes a lei out of flowers still inside their pots. Later, while listening to some records with hula music, Big Bird suggests that Snuffy can walk the 6,000 miles to Hawaii. But Snuffy says he's not allowed to cross the street alone, nevermind an ocean.
After catching the end of a game show in which travel destinations are awarded as prizes, everyone decides to participate in the next episode as a means to pay for Big Bird's way on their trip to Hawaii. They all dress in elaborate costumes (including Snuffy who appears disguised amongst the other contestants) which are judged by Guy Smiley. Big Bird catches the host's attention, not for his cowboy costume, but because he's assumed to be the large bird costume himself. After finding five things that begin with the letter B, Big Bird wins two tickets to Hawaii, thus covering Snuffy's expenses as well.
In an attempt to impress Bob with his preparedness, Big Bird invites him over to inspect how he's been packing for the trip to Hawaii. Bob and the kids help Big Bird decide what he should take and what he shouldn't; for example, an inner tube rather than a snow sled. Big Bird also asks them to help decide what he should pack for Snuffy: his giant toy sailboat or his oversized ice skates. Bob merely scoffs and ponders over the size of the items.
Everyone is getting ready for tomorrow's flight to Hawaii: Olivia makes a sweet potato pie to bring for Buffy and Sheldon, and Big Bird reminds Snuffy of the trip by dressing in a grass skirt. Meanwhile, Oscar is enjoying the rainy day they're having, but becomes disheartened when he learns that they won't be expecting poor weather during their trip.
Snuffy plans to follow the gang to Hawaii, but after Big Bird finishes packing, his friend misses the bus to the airport. Arriving at the terminal, everyone makes their way through the metal detector after an inconvenient delay getting Oscar's trash can through. Big Bird looks for Snuffy as he meets the pilot and learns about the plane just before everyone boards the flight. Snuffy arrives just in time to see the plane take off without him carrying Big Bird and Oscar who are worried about what they will eat on the long flight.
Buffy and Sheldon greet their friends who have arrived in Hawaii on Aloha Airlines. After receiving their leis, everyone boards a pickup truck for a ride out to the reservation where Buffy lives. On the way, they stop at a local fruit stand, Oscar gets cozy in Buffy's teepee, and the group meets some of Buffy's friends who arrive on a fire engine. Meanwhile, Snuffy has made his way to the island by way of helicopter and eventually meets up with Big Bird. Around the campfire that night, Big Bird hears the story of Mount Snuffleupagus, vowing the find the landmark with Snuffy during his trip.
The gang spends the day getting to know the locals: Oscar learns about native nationalities and asks about finding some lava, Olivia shares some poi with a new gentleman friend, Bob and Maria exchange songs in Hawaiian and English, and Snuffy emulates Keola playing a nose flute. Big Bird and Snuffy begin their search for Mount Snuffleupagus, while Mr. Hooper, Bob, and Olivia go canoeing.
The gang prepares for another trek, but Oscar attempts to procure his own mode of transportation. Big Bird and Snuffy strike out in their Mount Snuffleupagus search, but entertain themselves with hula dancing. After Bob and Mr. Hooper learn a traditional hula dance of their own, Bob asks Keola to teach him something romantic to recite for Linda before calling her back home and telling her he loves her. Bob later learns some Hawaiian games from the kids and Buffy introduces Big Bird to some of the flora and fauna on the island.
Determined to prove his age won't keep him from trying new and exciting things, Mr. Hooper learns how to search on the treacherous Hawaiian waves, an experience that prompts his immediate return to the ocean. Oscar learns what aloha means and gets a sea shell to squirt in Maria's face, while Snuffy uses a pay phone to call home to his mommy. Finally, after days of searching, Big Bird and Snuffy find the legendary Mount Snuffleupagus.
On their last full day in Hawaii, everyone prepares for a celebration of Cody's first birthday. The gang works in the garden, samples food for the party, and sing birthday songs in English and Hawaiian. At night, Buffy and Maria discuss their future plans for visiting one another before having to say goodbye the following morning. Boarding their flight, Big Bird is upset that Snuffy might miss the plane again, but once he's aboard and has taken his seat, he feels his friend's snuffle drop on his should from above; Snuffy has stowed away on the overhead compartment.
Back on Sesame Street, Susan, Gordon, Luis, Linda, and David hop on a bus to greet their friends at the airport. But, just as their bus pulls away, another arrives with their returning friends. Disappointed that no one has greeted them, the two groups eventually meet up with one another for a joyous reunion. Oscar gifts his neighbors some items he intended them to find yucchy, but they each find helpful uses for them. Back in his nest, Big Bird naps, surrounded by items he's brought back from his Hawaiian vacation.