At Lookout Landing, you meet Purah, who asks you to meet the search party at Hyrule Castle.
Find a Crystal Refinery
There is an exit from Lookout Landing beneath the platform where you spoke to Purah. Go out by this exit and go north to the destination.
Next to the path, you find some of the survey team examining an object that fell from the sky. If you climb up it and go around to the northeast side of it, you can talk to the Construct, who will tell you about the Crystal Refinery. However, you probably don't have enough materials to make Energy Wells yet.
Find a Korok
Go back down to the path that leads north from Lookout Landing, and go north beneath the stone archway. Use Ascend to go to the top of the archway and pick up the stone to find a Korok.
Then go back down and go north through the archway. You will see a shrine up ahead. As you approach the shrine, you will find Addison holding up a sign. If you talk to him, he will let go, and the sign will fall. You can help him out by attaching construction materials together to put under the sign so it won't fall when Addison lets go. For example, put a wooden board flat on the ground, and attach a short wooden board vertically to the middle, then attach three wheels on top of the vertical board, then place it under the sign. Talk to Addison and tell him to let go, and if you did it correctly, the sign won't fall. Addison will give you 20 rupees as a reward, and some food, and a Sleepover Ticket, which you can use at any stable. Keep an eye out for Addison elsewhere and you can help him out for more rewards.
Go to the Shrine
Now go to the nearby shrine north of here. This is the Kyononis Shrine. You can talk to Mylle, who is looking at the shrine, to learn more about chasms and shrines. Open the door of the shrine before you go, so you can teleport back here later. You can go into the shrine now if you wish, or return some other time.
Go to the Search Party
When you're ready to move on, look for the slope to the northwest. You can talk to Oliff there. Go north and you can use Ultrahand to open the gate. Beyond the gate, you can talk to Raseno. Continue following the path, and there is an entrance to the castle, where you can talk to Sharce, who won't let you pass until you talk to Hoz. Continue along the path, and you can talk to Yutas on the left side of the path. You can go into the door behind him to find a Traveler's Shield.
Go back to the path and continue following it. You can talk to Wingo, who is standing to the left of the path, and Flahva, who is near the edge of the chasm to the left.
In the gatehouse, you can talk to Purpen.
You can't go any farther in the gatehouse, and if you try to use Ascend inside of the gatehouse, you find that the ceilings are too high above you. However, you can stand in the entrance of the gatehouse, where the ceiling is much lower, and use Ascend there to reach the top of the gatehouse.
On top of the gatehouse, you can optionally stand within one of the windows and use Ascend to reach the very top, where you can climb to the top of the spire and examine the glowing object to find a Korok.
Back down on the lower part of the roof, talk to the soldiers there, and you will meet Hoz. There will be a cutscene.
Talk to Purah
If you opened the Kyononis Shrine earlier, you can teleport there now and go south from there to Lookout Landing. Otherwise, you will have to make your way south from here on foot (or horseback, if you can find a horse).
Talk to Purah and she will tell you to rest in the emergency shelter. If you haven't opened it yet, talk to Scorpis in the center of the village to open it, then go down the ladder and check on an empty bed to rest. If you need supplies, you can buy some from the shop in the east part of the town. If you need rupees, you can sell things like opals and rubies if you have found any.
Activate the Skyview Tower
After you rest, go talk to Purah at the Skyview Tower in town. After a cutscene, check on the terminal nearby. Purah will give you the Paraglider.
Now step onto the glowing spot inside of the tower and examine it. After the cutscene, your map in the Purah Pad will be updated with this area. After that, press the indicated button to glide. Be aware that you will run out of stamina after a while, but you can teleport back to the Lookout Landing Skyview Tower to avoid injury.
Talk to Purah Again
Now go back to Purah's lab and talk to her. She marks the locations of four places with strange phenomena. Josha mentions the Rito in Hebra in particular, but you can investigate these four locations in any order.
Josha Side Quest
Josha runs off after you talk to Purah. You can find her in the area beneath the platform where you talked to Purah. Talk to her, and during the cutscene, she gives you Brightbloom Seeds and Arrows. This starts the side quest Camera Work in the Depths. This side quest unlocks the camera in the Purah Pad, so you might want to do this side quest before you continue with the main quest.
When you are ready to proceed, you can go to any of the four locations that Purah marked on your map for the Regional Phenomena quest.