Main Quest: Locked Mementos
The Dueling Peaks Stable and Kakariko Bridge mark the fork in the road at which you can turn east instead of north to head out into the far eastern region of Hyrule and seek out Hateno Village.
The easiest way to get there is to fast travel to the Ha Dahamar Shrine. If you haven't completed the shrine yet, it doesn't take very long to get to the fork by foot or horse. You will soon find that your Sheikah Slate map is of very little help as you exit the Kakariko region. If you want to, you can fast travel to the Kakariko Tower and pin the Tower of this region and use that as an anchor.
You will eventually arrive at a Guardian graveyard along with a tall, imposing wall. This is Fort Hateno, a major landmark that marks the divide between areas ruined by Calamity Ganon and the remote settlements of the east that did not suffer as much damage. Feel free to scavenge the Guardians for their parts, as there are a lot of them, but beware the one Guardian furthest to the left that is still awake.
Beyond the fort is a sprawling wilderness. To immediate right is a cooking pot and a campfire, so use this as a chance to replenish supplies and take a rest if you need to. There are also several honeycombs that you can harvest. Use a Bomb to kill the bees first though, or they will attack you.
Note: There is a log cabin nearby the cooking pot. Talk to the professor inside, Dr. Calip, and you will unlock a Shrine Quest. Shrine Quests are sidequests that are necessary to undertake to unlock a specific shrine. In this case, the quest is The Cursed Statue.
The road will fork but both take you to the village. The south fork will let you pass by the Hateno Tower, this region's tower, so this is the way we are going. It is surrounded by Bokoblins and Moblins so approach carefully if you choose to do so. The Tower itself is covered in thorns so climb carefully around them. You can also set them on fire if you want to and use the Paraglider to catch the updraft and climb less.
Note: At this point in the game some NPCs have a chance of actually being members of the Yiga Clan, a splinter group from the Sheikah Clan that pledges allegiance to Ganon. Talking to them will cause them to reveal themselves and attack you!
Keep following the winding path straight and you will enter Hateno Village. There are multiple ways to enter the village. If you get stopped by a guard, they will eventually let you through. The NPCs you chat with will be more than happy to point you in the right direction to get to the lab. It is located in the outskirts of the village on a hill. There are also various shops to check out, including one shop where you can dye your armor different colors.
The Hateno Ancient Tech Lab that we are looking for is outside of the village proper, so make your way through all the business and you will find a long, winding path that takes you to a cape overlooking the Hateno Beach. The building at the very top of the cape is the lab. The path is marked by a bunch of curious lanterns so just follow them.
Enter and talk to the small girl standing in the middle of the room. Her name is Purah and she will tell you to go talk to the director. Go talk to Symin at the bookshelf. He will look at your runes and also tell you that actually, Purah is the director. She is just very brusque with visitors. Go back and speak to Purah again. She will help you fix your Sheikah Slate but only if you help her with the blue flame furnace.
Grab the Torch sitting in the Lab if you don't already have one and head back to Hateno Village.
Note: If it's raining you will not be able to do much other than wait for the rain to end.
The furnace is slightly outside of the village and is marked on your map. Wave your Torch at it to grab some of the blue flame. Now comes the tedious part.
You will have to walk with this flame all the way back to Hateno Village and light the torches you see along the way. When you finally reach the lab, light up what looks like a hot-air balloon to the left of the lab entrance. This will complete the task and also activate a small teleportation pad in front of the lab. Now you can fast travel to the Hateno Ancient Tech Lab anytime!
Note: If you read Purah's diary on the top floor she will prank you as punishment by pretending to delete all of your Sheikah Slate runes.
Report back to Purah. She will activate her Guidance Stone for you and your Sheikah Slate will be fully repaired. Now you have access to the Camera rune, which also will help you fill out the Hyrule Compedium.
To make sure everything is working, Purah will as that you take a photo of her. Press the L shoulder button and either use the motion controls or the right analog stick to make sure that Purah is in the focus of the image. Press [A] to snap the shutter and show the picture to Purah by talking to her again.
Purah will then tell you about the handful of landscape images that were also restored to your Sheikah Slate. She says that those were photos taken by Princess Zelda on her journey and that Impa might know more about them. As a final hint, she tells you that if you bring her Ancient materials she will give you "something good".
Note: After you are done talking to Purah you will unlock the sidequests Robbie's Research and Slated for Upgrades.
Return to Kakariko Village and go back to Impa to show her your newly repaired Sheikah Slate. She will indeed confirm that some of the images restored to your Sheikah Slate were taken by Zelda 100 years prior. This will lead to the main quest Captured Memories.
The Locked Mementos main quest is now complete. Next we will be journeying to the Divine Beasts, but you are free to seek out the locations shown in Zelda's photos.