In order to go on with your quest in another region, warp to the Lanayru Tower. Look at your surroundings, you should locate two towers and a shrine. Glide to the shrine towards the northeast, then keep following the path northeast on foot. When leaving the Lanayru region, you should see signs indicating Death Mountain and Akkala. Head left.
Keep going north and you soon reach the...
Talk to Gaile outside who tells you about Goron City and warns you against the heat. She suggests you buy some Fireproof Elixirs. Buy three of them. If you don't have enough rupees, you can sell some ore to Beedle. Take the Iron Sledgehammer if you don't have one. Talk to Mayro inside the tent who's looking at a painting. When you're ready, keep going east up to the...
Moa'Keet Shrine - Metal makes a Path
As usual, look at your surroundings. Use the Stasis Rune to freeze a rock ball, then hit it twice with the hammer to send it away. Run at the top of the slope, then go right. Use Magnesis to catch the magnetic ball, then climb the slope and drop it into the lava. Take the Knight's Broadsword into the chest, then after another magnetic ball has fallen down, go down the slope and go left. Use Magnesis again, grab another magnetic ball, then go forward after another rock ball has gone by. Then, still holding the magnetic ball in front of you, climb the slope with the four balls which will be repelled towards the top. At the top, drop the ball, look at the Sheikah symbol and the monk Moa'Keet gives you another Spirit Orb.
Go back to the stable, then join the path again towards northwest. After walking under some high rocks and that the path goes to the right, head left and start to climb towards the tower to the northwest. (You can also follow the path but it's very long and you're likely to meet a Guardian.)
Climb on some rocks, then do a little flight towards the tower, grab it and climb it to the summit. Place the Sheikah Slate on the terminal to get the map of the Eldin region. Now, remove your wooden weapon, bow and shield so they don't burn, drink one of your fireproof elixirs, fly towards northwest and reach the path. When you're next to Lake Goronbi, go south, where a shrine is located. Climb on some rocks and fly towards the shrine in the middle of the lake.
Qua Raym Shrine - A Balanced Approach
To start with, go ahead and notice the chest on a plate and some metallic crates at the back of the room. Use a Fire Arrow to burn the wooden crate on the other plate, which lowers the plate with the chest on it. Take the Small Key in it, then unlock the door, and take the three metallic crates out thanks to the Magnesis Rune. To get a Knight's Claymore, place a crate on the plate in the top-left corner (relative to the entrance) to make it go down, then another crate next to it, and from the stairs, jump on the crate and on the plate. Next, use the crates to make the plate near the elevated chest go down, then hover onto it, then again make the other plate go down to move up and hover to the chest. After getting its content, glide down to the ground, put a crate on the chest plate, then another one on the empty plate. Then jump on the chest plate and from there, place the third crate on the other plate. Place a bomb next to the cracked blocks, stand behind the crate to protect yourself and blow it up. Climb the steps, look at the Sheikah symbol and the monk Qua Raym gives you another Spirit Orb.
As you get out, drink another elixir, climb to the top of the peak and fly northwest. Reach the path, head west while avoiding the molten rocks, cross a small bridge and you arrive in the...
Talk to the Gorons and to the explorer Kima if you want to, but don't stay too long because of the heat. The worker Tonjaga tells you about his boss and Goron City. However, if you are about to run out of fireproof elixir, you can look for 10 Fireproof Lizards nearby and give them to Kima, who will hand a protective armour over to you.
Carry on your way to the north. Beware of the falling burning rocks, then a cutscene showing a giant lizard-like creature starts. Walk some more to reach the entrance of...