After meeting with Impa in Kakariko, she tells you to visit the Hateno Ancient Tech Lab.
Activate Hateno Tower
After you are done in Kakariko, you should activate Hateno Tower.
Go to Hateno Village
From Hateno Tower, you should follow the road east to reach Hateno Village. There is a shrine within the village. It is the Myahm Agana Shrine. Open it to create a warp point here. You can also complete the trial in the shrine if you wish.
Go to the Hateno Ancient Tech Lab
Follow the path east through the village, and go to the building at the top of the hill east of the village. Inside, talk to Symin, then Purah. Ask her to restore the functionality of your Sheikah Slate. Purah asks you to light the ancient furnace with the blue flame.
Light the Ancient Furnace
When you go outside from the lab, look toward Hateno Village. You should see a bright blue light to the right of the village. That is the town's ancient furnace. You will need a torch, and you can get one in the lab if you don't have one. Then glide down to the blue flame, equip a torch, and swing it at the blue flame to light your torch. Remember not to run while carrying the torch, because you will blow out the torch and put it away if you try to run.
There is a stone lantern nearby. Swing your torch at it to light it with the blue flame. There are more stone lanterns down the path toward the village. These stone lanterns stay permanently lit, so if your torch goes out, you can re-light the torch later with one of the stone lanterns instead of having to go all the way back to the original blue flame. The lanterns are all along the path through the village and along the path to the lab, so if you are having trouble finding a lantern, stop and look around the main paths.
Get New Sheikah Slate Features
After you have lit the ancient furnace in front of the lab, you will be able to warp back to the lab at any time. Go in and talk to Purah. You will use the guidance stone. This adds the Camera Rune to your Sheikah Slate, as well as some photos, and the Hyrule Compendium. Take a picture of Purah with the camera, then talk to her to show her the picture. Purah says to go talk to Impa. Purah also wants some ancient materials, and she says that Robbie in the Akkala Ancient Tech Lab also wants some ancient materials. This adds the Robbie's Research and Slated for Upgrades quests to your Adventure Log.
The Hyrule Compendium will record information about things that you take pictures of around Hyrule. Only certain types of things can be recorded in the Hyrule Compendium. For example, you can't record monster parts in the compendium.
The Hyrule Compendium is especially useful if you ask Purah to upgrade your Sheikah Sensor. If you have been searching the dead guardians around Hyrule and defeating guardians in shrines, you should have enough ancient materials to get the Sheikah Sensor upgrade from Purah. When you get the upgrade, talk to Symin to start the Sunshroom Sensing quest. The Sheikah Sensor upgrade lets you sense items that you have taken pictures of in the Hyrule Compendium. Just open a Hyrule Compendium entry and press A to select it for the Sheikah Sensor. You can switch back to shrine sensing if you go to your map and press Y. Be sure to take a picture of a treasure chest for the compendium so you can use your upgraded Sheikah Sensor to find treasure throughout Hyrule.
After you get the Sheikah Sensor upgrade, talk to Symin to start the Sunshroom Sensing quest. Go out through the door near Symin. There is a Sunshroom nearby. Take a picture of it with your Camera rune. It will be added to the Hyrule Compendium. Go talk to Symin and tell him you took the picture. Then he asks you to collect three Sunshrooms for him. He says that they can be found in Retsam Forest, which is just north of the lab. Press - to open your Sheikah Slate, then press R to go to the Hyrule Compendium. Select the entry for Sunshroom, then press A to target it with the Sheikah Sensor. Then glide down to the forest and collect the Sunshrooms. Then you can warp back to the lab and talk to Symin to complete the quest. He gives you three Hearty Truffles in return. If you cook one, it creates a meal that fully recovers your hearts.
Bonus Compendium Pictures
You can talk to Symin again to get some new pictures added to your Hyrule Compendium. From now on, you can talk to Symin to pay to add a random entry to your Hyrule Compendium. This is helpful if some entries are too difficult to get, or if you missed a one-time-only item.
Talk to Impa
Now that you have repaired the Sheikah Slate, warp to Kakariko Village. After you talk to Impa, the Locked Mementos quest is completed, and the Captured Memories quest begins. Impa says to visit her after you have recalled one of the memories from the pictures, because she has something to give you. It is a very useful item, so you should do this soon.
In Kakariko, you can find a guy named Pikango standing near the goddess statue across from Impa's house. Talk to him and say that you understand, and say that you want to go along with him while he looks for the Great Fairy Fountain. This starts the "Find the Fairy Fountain" quest. Follow him up the hill. He will go up the slope, past the shrine, and then he will stop.
Go farther along the path, and it will split. There is a lower path on the left, and an upward slope on the right. Go to the left. You will find a strange plant with bright orange mushrooms in front. There are fairies floating around, so walk slowly to avoid scaring them away, and take a picture of one for your Hyrule Compendium. Then carefully collect the fairies.
Finally, walk onto the bright orange mushrooms and press A to examine the giant plant. The Great Fairy Cotera asks you for 100 rupees. If you don't have them, go into town and sell stuff until you do, then come back here and give her the rupees. Then she can increase the defense of your clothing for free if you have the required materials. If you don't have the required materials, you can talk to her to find out what you need.
After you have opened the Great Fairy Fountain, you can take a picture of it for Pikango (it doesn't work to take a picture of it before you have given the Great Fairy the rupees). After you take the picture, go show it to Pikango. After that, he will look at the pictures in your Sheikah Slate that you need to find to recover your memories, and he will tell you that one of the pictures is Mount Lanayru, and the photo was taken just west of Naydra Snowfield.
After this, you can find Pikango at certain stables and other places, and he can tell you the locations of other pictures.
Recall a Memory
The Memory Locations page has instructions for how to recover each of Link's memories.
Pikango mentioned that one of the images from the Sheikah Slate is of Mount Lanayru. Mount Lanayru is already on your map if you activated Hateno Tower on the way to the Hateno Ancient Tech Lab. Mount Lanayru is east of Kakariko Village, so after fulfilling Pikango's request, you can go east from Ta'loh Naeg Shrine to recall the memory from the picture. However, you will have to fight Bokoblins and Moblins to get there. If you want to find an easier memory, look at the second picture in the top row. You should be able to recognize the Dueling Peaks. You should be able to find the location if you go toward Central Tower.
After recalling a memory, don't forget to talk to Impa to get your reward.
At this point, you can try to go to the locations marked in the Free the Divine Beasts quest in your Adventure Log. For guidance with the Free the Divine Beasts quests, read Reach Zora's Domain, Reach Gerudo Town, Reach Rito Village, and Reach Goron City.
You can also try to find the locations in the pictures to recall your memories, or try the side quests that you have been given, and talk to people throughout Hyrule to get more quests. Or, as always, you can just explore.