After you receive guidance from the past, you seek the Master Sword.
Talk to Purah
After the events in the Spirit Temple, return to Lookout Landing and talk to Purah. This completes Find the Fifth Sage, and Crisis at Hyrule Castle.
Talk to Hestu
Hestu is in Lookout Landing, between the stable and the shop. Talk to him to learn that something is not right in the Korok Forest.
Go to the Great Hyrule Forest
To reach the Great Hyrule Forest, go east from Lookout Landing and go straight at the first fork in the road, then go left at the next fork, so that you are going northeast. You will cross Rebonae Bridge, and you can optionally go south to Wetland Stable, and visit Tukarok Shrine nearby.
Take the road north from there, then turn right at the fork. When the path splits, go straight, and you will reach Thims Bridge. After that, go left at the fork in the road so you are going northwest. You will reach Woodland Stable, and you can visit Ekochiu Shrine nearby.
From here, you can go north to reach Great Hyrule Forest.
Lost Woods
When you enter the Lost Woods, you find Peeks near the entrance. He says that there is no way to get past the fog here, but a hole opened up in the Minshi Woods.
You went past the chasm by the road on your way here. Go southeast and go south along the road. You will find a Korok named Tasho standing next to the chasm. He says his friends jumped into the hole, hoping to find a way back to the Lost Woods. To proceed, fall into the hole.
Minshi Woods Chasm
In the area where you land, you can talk to a Korok named Mopu, who points out another Korok. Go northwest to talk to Nako the Korok. You will see a Lightroot to the northwest. Summon Mineru's Construct and ride on it northwest toward that Lightroot.
On the way, you can talk to Mokka the Korok, who points out the lost fog, above which Mokka says the forest should be.
Instead of going northwest to the nearest lightroot, go west. There are gloom monsters in that direction that can easily knock you off of the Construct and grab it.
Go toward the brightly lit area and you will find a lighted path. It leads to a flat platform. Activate the Lightroot near this platform. You can talk to the Koroks nearby to learn that the Great Deku Tree would be directly above you. Stand in the middle of the platform and use Ascend.
When you reach the top and exit the ground, the Deku Tree will call to you. First, open the door of the nearby shrine. Then walk up the root in front of the Deku Tree, stand on the platform with the small branch sticking up out of it, and talk to the Deku Tree.
Then go down below and enter the room beneath where you spoke to the Deku Tree. In the back, you find a chasm. Before you go, you might want to talk to the Bargainer Statue near Josha in Lookout Landing to purchase the location of one of its brethren statues, then go to the marked statue to buy a Tunic of the Depths, which provides some gloom resistance.
When you are ready, fall into the chasm in the Deku Tree.
Deku Tree Chasm
You will immediately be attacked by gloom monsters. When they are defeated, you have to battle Phantom Ganon.
Afterward, go back up to the Deku Tree, either by teleporting to the shrine or by using Ascend.
Walk up the root in front of the Deku Tree, stand on the platform with the small branch sticking up out of it, and talk to the Deku Tree. You will learn the whereabouts of the Master Sword. This completes the quest Trail of the Master Sword and begins the quest Recovering the Hero's Sword.