I found the blue flame furnace and lit the torches, but when I got to the Hateno tech lab, the thing to light the blue flame into was closed. Symin and Purah have different dialog if you haven't talked to Impa yet, and Purah's diary isn't in her room... So it turns out the "Seek out Impa" quest is required before you can get any of the Shiekah Slate upgrades or the camera. I bought the soldier armor here cuz I know it has the most defense out of all the armor in the game.
From there I decided to go to that one shrine on the cliff in the distance that I can see from the tech lab... but that cliff is filled with really tough enemies... I had to improvise with my deku leaf and bombs and just barely managed to get by. It was a Modest Test of Strength shrine, and beating it used up all of my good weapons.
Then I went to the shrine out in the ocean, and it was a Major Test of Strength so I was like fuck that, and then I went to Eventide Island instead... I spent way too long attempting it, but kept getting killed by random shit... even just the normal enemies around the island are a huge ordeal, and the only way I can think of to kill them is to use bombs to knock them away until they fall off a cliff or into the water. But then I found out there's a black moblin on the island and I can't kill him that way... eventually I was like fuck this, I can't do it yet until I get more hearts.
I went back to Lanayru Tower... and was thinking of maybe doing the Zora quest now, but then realized I wanted to do more shrines instead. I went and beelined towards the 2 shrines in Lanayru Wetlands... and then went to upgrade my armor and spent like all of my ore money on the stealth armor set.
Then I warped back to the Plateau in order to start exploring south Hyrule... I think the plan now is to do the divine beasts in clockwise order starting with Gerudo and ending with Zora, and probably get the Master Sword along the way, and then finally after all that, talk to Impa and see what happens if you've already done all the beasts before you complete the "Seek out Impa" quest. I went to the shrine in the middle of Lake Hylia, then beelined toward Lake Tower. This tower was very well guarded by tough enemies.
From there, I looked around and wondered if I'd be able to get into the desert from the southern highland cliffs... and went towards a shrine and Highland Stable. This place has Super Rumor Mill EX V3, which tells you where to get the DLC horse equipment. Oh and I went to the Horse God fairy at some point during this to get the ancent saddle... it's on the main path and very easy to find.
And then I saw the Faron Tower so I tried to head towards that. And also got distracted by that shrine behind the waterfall on the way there... It felt like a really neat challenge, getting up to that tower without any stamina upgrades. It made me go a certain way in order to find the easiest climbing opportunities... and I also had to fight through a lot of Lizalfos and some thunder Wizrobes. By now I was like, way stronger than when I first started playing today, although it's still tough. I'm still at the level where I'm often trading 1 or 2 weapons for 1 weapon whenever I defeat an enemy.
Finally after getting Faron Tower, I went to explore Faron Grasslands and the Gerudo Highland cliffs... I got to the "You can't go any further" point at the south, where it's just wasteland off in the distance (I guess the ocean shoreline curves around somewhere beyond the southern Faron border)...
I kept looking around, and eventually found a way up into the highland, and got through a snowy area where I was taking damage even with the warm doublet and cold resistance food, and then I ended up at the South Lomei Labyrinth. This didn't take me long to do at all, and got some really good weapons in there too, and explored the "Champion's Gate" area which I don't even remember ever exploring in my first playthrough. It's kinda like a rocky hill between the desert and the highlands.
From there, I went towards the Wasteland tower and stable, and stopped there... The scenery in the desert/wasteland area is absolutely amazing. By now, I'm definitely on pace to have enough spirit orbs for the Master Sword by the time I finish getting all the towers and easy shrines.