From Bulbapedia, the community-driven Pokémon encyclopedia.
Via an event at the Spinda Café, you'll unlock Shaymin Village. The only purpose for coming here is to get a Gracidea flower, from a Shaymin next to the Kangaskhan Rock. After this, head up into the Sky Peak Stations, which is above the Shaymin Village!
Sky Peak Station Passes
Since this is your first time going through these dungeons, a Shaymin will be with you. Since it's a legendary in the main series, this makes you have a lot of power, so this shouldn't be as hard as it would be.
After finishing 3 or 5 of these, you'll reach clearings. These clearings don't do much except give you a Kangaskhan Rock. However, on certain clearings, there will be a base that a Breloom, Machoke, and Mawile set up. All three of those Pokémon are part of Team Frontier.
Once you reach the fourth pass, Team Frontier will confront you and say that an explorer called Sneasel got confronted by a group of unknown Pokémon. Team Frontier, worried for Sneasel's well being, hurry off. However, these "unknown Pokémon" are not so unknown to Shaymin--since it rushes you to get ready and head through the fifth pass!
At the end of the dungeon, you'll see Sneasel being harassed by those unknown Pokémon. Team Frontier is getting upset and decides to battle them. Meaning, they'll help you battle them as well.
At the fifth pass, on your first time going through these dungeons, there will be a boss (or, even more fitting, bosses). In this case, it's five Carnivine. As they are Grass-type, Fire, Flying, and Ice moves could work, however if you have any of the other types it's even better. After defeating them, they'll let you through. Shaymin explains that they're really nice Pokémon, but can get tipsy when someone else steps onto their territory.
On the Seventh Clearing
You'll see Team Frontier, surrounding Sneasel. Your team quickly rushes forward, wondering what happened. Breloom says he doesn't know, as Sneasel was knocked out before they came here, not giving you much information. Breloom wonders if it's because of the cold or because he ran out of items. Shaymin says it'll go get help, but Mawile stops it, asking who. Shaymin explains there's a rescue expert at the eighth clearing. It asks Team Frontier to look after Sneasel as your team goes ahead.
On the Eighth Clearing
When you finish the eighth pass, you'll find an Ampharos, walking around. Ampharos starts a cozy talk with Shaymin, but Shaymin, though enjoys the small talk, says that there is an emergency. Shaymin hurriedly explains that the problem is on the seventh clearing. Ampharos listens to Shaymin and quickly goes.
It shows Team Frontier, mourning over Sneasel's "death". They get anxious when Ampharos comes, but Ampharos says it's here to help, and decides to take Sneasel to the eighth station clearing.
Sneasel wakes up a while later, on Ampharos's bed. Shaymin is concerned and asks if he's alright. Sneasel doesn't here Shaymin and wonders where he is. Mawile begins to explain that Sneasel passed out on the seventh clearing. Breloom indirectly thanks Ampharos for its help, as if Ampharos didn't come, then Sneasel would not be alive right now. Sneasel is surprised that anyone would ever do that for him, but Ampharos explains it wasn't just it--it starts to be modest, giving credit to everyone. Though Sneasel attempts to thank you, Sneasel falls back over, but Shaymin knows his words of thanks, saying Sneasel needs to rest. Ampharos gives you permission to go on to the summit, as it is going to take care of Sneasel itself.
The Pokémon that can be found here begin with Pokémon found in the Mystifying Forest, however in later floors there are more Pokémon from Mt. Travail, a dungeon found later in the game.
Sky Peak Summit Pass
This dungeon, being the one after all of the Station Passes, is considerably harder than other ones, however since you have Shaymin with you it should be a lot easier. There are 6 floors here, in which 2-5 floors can get either Monster houses, hidden stairs, and the Secret Bazaar can possibly make it here.
You make it to the summit. It's very foggy, and it looks like some smog is floating around. Apparently, it happened because Shaymin hasn't been to the summit for so long. Suddenly, Team Frontier comes! They are also surprised at the sudden fog. It's so different, Mawile can't even believe it's the summit. Without warning, some voice says their love for dust, dirt, and slime.
Suddenly, five Grimer and three Muk come out! Shaymin is annoyed by this and tells the Grimer and Muk that they don't belong here, and asks nicely for them to go back home. The Grimer strongly disagree, unfortunately. One of them even thinks we're lying! The Muk says that they hate "cleanliness" and drive it away. Shaymin grudgingly says they'll have to fight them...
Since this is with Team Frontier, this shouldn't be as hard as it would be if it was just your team and Shaymin. Although all of them can use Minimize, Mawile can solve that problem with Sweet Scent, so this can be one of the easiest battles in the game.
All of the Grimer and Muk yell random things, and extend the words they say. Shaymin asks if they have "come to their senses". The Muk and Grimer wonder what Shaymin means, though soon find out and are extremely surprised. Shaymin asks what happened when it was away for so long. Shaymin tells them that this isn't their home. The Grimer are surprised to hear that and protest, as it's dirty and grimy. Shaymin then finds out...
No one's been there for so long. Shaymin apologizes, and says that this is the mountain summit, not their home. The Grimer appear to be very scared of the summit. Shaymin thinks it fell into disrepair since no one's visited for so long. Shaymin, though feels bad for the Grimer and Muk, refuses to let the beautiful summit come to an end just because of the grime, and asks them to return to their rightful home. The Grimer and Muk then go away.
Shaymin wants to clean up the place, and restore it back to full health. Mawile doubts this can happen, though Shaymin explains all Shaymin have a special power to clean up land and absorb dirty stuff. All the same, this is no easy task. Shaymin begins the task and tells you to get down, as it continues to store seeming dirty energy. It then releases all the energy.
The truth is the summit is a beautiful garden with many flowers. Mawile is surprised this is the summit. It's so beautiful... Machoke has nothing to say, though it might not be because the flowers literally took his breath away. Shaymin congratulates everyone on reaching the summit. Machoke is very happy to have made it. Mawile notices something and tells everyone to look over somewhere. It's the beautiful view of mountains seemingly sitting on the clouds. Breloom thanks Shaymin for taking all of you there. Shaymin says they should go back down, and everyone agrees.
...Until they see the downside as they wonder how to get back down. Shaymin does not worry and says it'll take all of you back to the base of the mountain. Mawile wonders how such a small body can carry this much Pokémon, but Shaymin says the flowers here are Gracideas, a flower that's special for the Shaymin. Shaymin touches the Gracidea and pow...changes forme! Everyone is surprised, but Shaymin disregards this. Breloom wonders if Shaymin is adorable...(what?!) while Machoke is speechless. Shaymin doesn't care about that either and says that in this forme, it can fly. All of you then go to the base of the mountain, hanging onto Shaymin.
Continue to Section 2.