After you meet Volcanion in Rocky Ridges and Raikou in Bleak Beach, you can proceed through the north area gate to Sparkling Skylands.
How to Reach Sparkling Skylands
After you meet Volcanion in Rocky Ridges and Raikou in Bleak Beach, go through the north area gate to reach Sparkling Skylands.
Follow the Professor
Go up the stairs and talk to the professor. Then follow the professor to the tower and talk to them.
Get on the Platform
Step onto the metal platform and press A. You will ride the lift.
Talk to the Pokémon
Go over to the Pokémon Center and talk to the Pokémon to meet Rookidee.
Check the Pokémon Center PC
Go check on the Pokémon Center PC.
Raise the Environment Level to 3
You and Rookidee will start out without a home, so you might want to build one and put a Ditto flag there. You can build homes (Rookidee will soon ask you to use a building kit to make a house) and restore habitats to make more Pokémon move in. Improve the habitats and give Pokémon their favorite foods to help improve the environment level.
Rebuild the Pokémon Center
After you raise the environment level to 3, Professor Tangrowth will ask you to rebuild the Pokémon Center. You can buy the Pokémon Center Rebuilding Kit from the PC shop. The professor will also teach you the Waterside Dinghy habitat if you don't know it already.
Learn to Fly
Build the Waterside Dinghy habitat (see above) by using Leafage in the water where there is a canoe. (Using Leafage while standing in water will create duckweed in the water.) When you meet Dragonite, they will teach you how to fly. Follow them and talk to them to proceed with their request.
Meet Tinkmaster
On the distant island, Dragonite points out a sound coming from the nearby tower. Go up and you will find a Poké Ball that contains Iron Ladders, so use them to climb the rest of the way up. Then you meet Tinkmaster. If you don't yet know the Construction-Site Generator habitat, you will learn it now.
Get Limestone
Before you go back to the previous island, go southeast from the tower to find a bunch of Limestone. Gather some, you can only make concrete if you have limestone.
Talk to Conkeldurr
Back on the first island, there is a furnace next to a metal tower. In the tower, you will find iron scaffolds and iron pipes. Move them next to the furnace to build the Construction-Site Generator habitat. When Conkeldurr moves in, talk to them. You will learn the Concrete Mixer recipe. Make one and put it on the stone tiles next to the stream near the furnace. If you haven't already, go up and break the rocks that are blocking the water above. When the concrete mixer is in place next to the water, put limestone into it, then have Conkeldurr follow you to it.
Talk to Tinkmaster
When you have six bricks of concrete, talk to Tinkmaster. You will need one Iron Ingot, so get some iron ore (there are scraps of it lying around, and you can break rusted stairs and other things to get them) and put it in a smelting furnace, and bring a Pokémon with the Burn specialty to the furnace to make at least one iron ingot. Then talk to Tinkmaster when you have it. Wait for Tinkmaster to be done (or come back later), and you'll get Tinkagears.
Follow Tinkmaster
After you get the Tinkagears, follow Tinkmaster to the platform where they want you to put a lift platform. You will learn the Lift Platform recipe. There is a workbench nearby that you can use if you already have the materials. Once you have made a Lift Platform, put it where Tinkmaster showed you, making sure that the rail is pointing toward the Lift Platform on the other island.
Escort Tinkmaster
Talk to Tinkmaster and agree to take them with you, then use the lift platform that you put down, and Tinkmaster will use it as well. Go to the Pokémon Center with Tinkmaster.
Optional: Give Tinkmaster a Home
At first, Tinkmaster has no home, so you might want to give them one since it will help improve the environment level.
Talk to the Professor
Talk to the Professor, then follow Tinkmaster, then talk to them. Build a road to the island with the tall building. After you go up the asphalt, turn right and go up the stairs, and follow them up to the building. Approach the building and talk to Tinkmaster. Then follow Tinkmaster and talk to them when they stop. Continue to follow them and talk to them, then follow them back down to in front of the building. Talk to them and they will set up a building kit in front of the building. You can now add materials and bring Pokémon to help rebuild the building.
Rebuild the Building
Bring materials and Pokémon like usual. When 2F is finished, you will need to bring Chef Dente from Rocky Ridges to work on 3F. Go talk to Chef Dente, then use the northern area gate to go to Sparkling Skylands, and take Chef Dente to the building site.
When 3F is finished, you will need to bring Peakychu from Bleak Beach. Talk to Peakychu, then use the northern area gate to go to Sparkling Skylands, and take Peakychu to the building site. You will also learn the Professor's Apprentice Program habitat so you can meet Metang, who can recycle wastepaper into paper, which is needed to repair 4F.
Go to the Roof
After 4F is repaired, the building is complete. Go to the roof (you can either use the elevator inside the building or the stairs around the back) and check on the floating Master Ball to meet Mewtwo. Talk to them to proceed. Your Trainer Rank will increase to Master.
Talk to the Professor Again
After you meet Mewtwo, go back to the ground floor of the building and talk to the Professor to complete the important request.
Raise the Environment Level to 5
If you haven't already, raise the environment level to 5, then talk to the Professor until you get a Game Boy System from the PC.
Complete the Team Initiation Challenge
You can now complete all stages of the Team Initiation Challenge.