Impa says to seek the Goron elder to free Vah Rudania, so you should go to Goron City.
To Lanayru Tower from Kakariko
If you have been to Lanayru Tower or the Soh Kofi Shrine near it, warp there and skip to the next section. If you have not yet been to the Lanayru Tower, warp to the Ta'loh Naeg Shrine in Kakariko Village, then go northeast, and you will reach a fork in the path where the right side slopes upward, and the left side goes straight ahead. Go left at that fork and continue going northeast.
At the next fork, go north until you see Lanayru Tower. If you go straight toward the tower, you will soon find the Rutala River blocking your way. However, you can float down to an island that sticks out from the opposite side of the river. Go northwest along this island, then go northeast from the sleeping Hinox and cross the bridge. A Zora may speak to you here. You can read the Reach Zora's Domain page if you want to go to Zora's Domain from here instead of continuing on to Goron City. Otherwise, go east on the bridge. From here, you can make your way up the slope to the Lanayru Tower.
North from Lanayru Tower
From Lanayru Tower or the nearby Soh Kofi Shrine, go north to a path that goes northeast. Follow this path, going left at the fork (the left side of the sign says Maw of Death Mountain). Soon after you go left, you will see a shrine in the distance. Keep following the path and you will reach Foothill Stable. Go east from the stable to reach Mo'a Keet Shrine.
Mo'a Keet Shrine: Metal Makes a Path
There are slopes where boulders are rolling down. Before going up the first slope, go past it and look at the ceiling. If you have bomb arrows, you can shoot the cracked rocks in the ceiling to get a chest. If not, there is another way that you can break the cracked rocks.
Go up the first slope, using Stasis to stop the oncoming boulder if necessary. Try to stop the boulder as soon as it comes down from above. Otherwise, it might block your path.
At the top of the slope, go to the right, where you will find some magnetic boulders rolling down. Use Magnesis to stop the boulder, then go up the slope that it came down and you will find another chest. You can take the magnetic boulder back down to the lower level where you entered the shrine (using the magnetic boulder to block the non-magnetic one in the middle as you go down the ramp) and use the magnetic boulder to break the cracked rocks in the ceiling to get the chest there.
Take the magnetic boulder to the ramp on the left where a lot of boulders are coming down. Use the magnetic boulder to block the other boulders so you can safely go up the ramp. You may need to press up to push the magnetic boulder forward through the other boulders.
At the top of the ramp, you can get the Shrine Orb from Mo'a Keet.
To the Next Tower
Back outside, go west to the stable. You can buy Fireproof Elixirs from Gaile at the stable, or you can wait and get the ingredients for them yourself soon. When you are ready, go north on the path from the stable. You should be able to see the tower straight ahead.
Look for Fireproof Lizards in this area. You can use them at stable cook fire to make Fireproof Elixir (use 1 or more Fireproof Lizards and 1 monster part). You will need the elixir to reach Goron City.
Go north from the stable again to reach the Maw of Death Mountain area, where there is a hot spring surrounding a rock path. If you get in the spring, your hearts will slowly be restored.
After going past the hot spring, there will be a Guardian Stalker on the right. You can go left from here to climb up the volcano wall and reach the tower quickly. The path with the Guardian Stalker doesn't have anything special on it (except for another Guardian Stalker farther up the path, and some areas of extreme heat) so climbing the volcano wall is recommended.
From Eldin Tower to Goron City
On your map, you can see some small hot springs northwest of the tower. Go to these hot springs instead of following the main path. This allows you to avoid some enemies along the main path, including an Igneo Talus miniboss.
Northwest of the hot springs, you reach Gortram Cliff. To the northwest, there is a bridge going over a narrow span of lava. Go down to that bridge and follow the path. This is the Southern Mine area. Look for a Hylian in this area named Kima. Talk to him, and if you give him 10 fireproof lizards, he will give you Flamebreaker Armor, which will protect you from the extreme heat in this area.
Go north from here and continue along the path. It leads to Goron City.
In Goron City, go to the right from the entrance to go up a slope that leads to the armor shop. You will need at least two types of Flamebreaker gear to survive the extreme temperatures as you get closer to the Divine Beast in this region, so be sure to buy some armor here.
Then go to the destination marked on your map in this area for the Free the Divine Beasts quest, and you will find the elder, Bludo. When you talk to him, he will mention Yunobo, who hasn't yet returned from the Abandoned North Mine. This starts the Divine Beast Vah Rudania quest.
Go north along the path from the elder. Eventually you reach the Shae Mo'sah Shrine.
Shae Mo'sah Shrine: Swinging Flames
Up ahead, there are dry leaves on the ground and walls. If you don't have fire arrows or another source of fire, you can shoot an arrow at chain holding the flame ball above the hole in the ceiling. When you burn the leaves, the stone barrel will fall. Put it on the switch to open the bars. Go through.
There is a treasure chest to the left. Across from the chest, there is a gear on the wall. To the right of the gear, there are a few Guardian Scout I. You can defeat them by doing a perfect guard when they shoot a beam at you, or just hit them with your weapons until they die.
In a corner of the area with the Guardian Scouts, there is a treasure chest. In another corner, there is a ramp up to the next area. Go up that ramp. You will see some bars nearby, but you can go away from those bars to find stairs up to a glowing orange switch. Step on it to make a flame ball drop from above. Use Magnesis to pull the ball toward you, then let go to make the ball hit the leaves. The leaves will burn the wooden platform, making the small metal ball roll down until it stops at some bars.
Before you continue, drop down to the area below where the flame ball was hanging. There is a treasure chest there.
From the switch that you stepped on to make the flame ball fall, go along the raised path that goes alongside the ramp that the metal ball is on. Go all the way to the wall with the constellation on it, and go left to find a switch. Step on it to open the bars, allowing the metal ball to roll and rotate the gear. This opens some nearby bars. Go through to find a treasure chest that contains a small key. Then go up the ramp to leave this caged area.
Go back to the wall of bars with the locked door, and unlock it. Step on the orange switch to make two flame balls drop down. Use Magnesis to make one of them swing back and forth, then shoot an arrow at the chain when the ball is moving away from you. It will hit the burning leaves, making the stone barrel fall. Put the barrel on the orange switch and go through the door to reach Shae Mo'sah.
Abandoned North Mine
Back outside, go north from the Shae Mo'sah Shrine to reach the Abandoned North Mine area. You can kill the Rock Octoroks by throwing a bomb at them when they are inhaling air. The Octorok will swallow the bomb, which you can then detonate to kill the Octorok.
Be careful of volcanic eruptions as you go. You might need to hide under a ledge to stay safe.
In the Abandoned North Mine area, you can talk to the Goron to find out where Yunobo went. In this area, you will find a cannon. The Goron will tell you not to use it, but you can ignore him. You can drop a spherical bomb into the cannon tube, then detonate it to launch the bomb and make it explode where the cannon is aiming. You can change the cannon's direction by hitting the lever on the back of the cannon.
Hit the lever to make the cannon point at the left, and press L to detonate the bomb to shoot the Lizalfos there.
Make sure you are equipped with at least two pieces of Flamebreaker gear, then glide down to where you blew up the Lizalfos. There is another cannon there. Hit the lever to make it point at the two Lizalfos farther along the path, and put a round bomb in the cannon and detonate it to kill those two Lizalfos.
Go to where those Lizalfos were. There is an updraft just past where they were. Glide onto the updraft, ignoring the Lizalfos that shoot fire arrows at you. Keep gliding on the next two updrafts and you will reach another cannon. Use it to shoot at the Lizalfos who are shooting fire arrows. Afterward, rotate the cannon and shoot a bomb to kill another Lizalfos shooting fire arrows.
You can also shoot the pile of cracked rocks on a pillar between the two sets of Lizalfos that you bombed. To hit those rocks, you will have to detonate the bomb while the cannon is moving.
Then go up the slope that goes in the direction of that stone pillar that had the cracked rocks on top. Glide onto the updraft to the right, and ride it to the top of the stone pillar, where you can hit some ore deposits. From there, float on another updraft to reach the next section of rock.
Go north along the rock path until you reach another updraft. Float in it to the second updraft, and float on the second updraft to reach another cannon. You can use the cannon to kill several Lizalfos in the distance, including one next to a stone skull.
After killing the Lizalfos, go toward where they were. There is a cannon to the left that you can use to destroy the platform that some more Lizalfos are standing on. There is another cannon beyond that platform. You can float on the updrafts to reach it. From here, you can see the structure where Yunobo is trapped. Use the cannon to break the cracked rocks that block the entrance to that structure, then float there and talk to Yunobo.
After the cut scene, warp back to Goron City and talk to Bludo the elder. After the cut scene, you should go to Eldin Bridge. It's southeast from Goron City. One way to get there is to go up to the Shae Mo'sah Shrine that overlooks Goron City, climb the wall across from the shrine, and use Magnesis to lift the mine cart onto the track up above. Make sure that the lantern on the cart is facing in the direction of the tracks. Then put a round bomb in the rear compartment of the mine cart and detonate it. It can be difficult to keep your footing, so make sure you are standing safely on the cart as it moves.
Get out of the cart at Stolock Bridge. Cross the bridge. You can use the mine cart at the other end if you want to avoid some red Chuchus.
Continue going along the path. You will see some hot springs up ahead. You can use Magnesis to put the mine cart on the track, and use bombs in the mine cart's rear compartment to ride in the cart and avoid more monsters up ahead.
At the end of the mine cart track, you will see Daqa Koh Shrine up ahead.
Daqa Koh Shrine: Stalled Flight
Get on the rising and falling platforms in the middle of the room. Start gliding when you reach the highest point, then land on the big stone that has an electric crystal on it.
If you want to get the treasure chest in the shrine, turn around and face the elevator that you entered from, and you will see a treasure chest on a ledge. You need to be standing on the stone with the crystal on it to be high enough to float down to the treasure chest.
To complete the shrine, get on top of the stone with the electric crystal again. Switch to the Stasis Rune, then when the crystal reaches the top, use Stasis on the stone under the crystal to freeze it in mid-air. This will keep the bars open so you can float down to Daqa Koh and get the Shrine Orb.
Reach the Divine Beast
East of the shrine, you will find the Bridge of Eldin. Kill the two Moblins who are harassing Yunobo, then talk to Yunobo. He will get in the cannon. Hit the lever to rotate the cannon so it aims at the Bridge of Eldin, then put a bomb into the cannon and detonate it. The bridge will fall. Cross the bridge, and Yunobo will follow you.
The Divine Beast sends out Guardian Sentries. Tell Yunobo that you can whistle to signal to him.
Make your way up the path, and try to reach the large rock outcropping without being seen. Then when the sentry is going back the way you came from, go forward along the path with Yunobo.
Soon you will encounter a Sentry that is not moving. Whistle to make Yunobo stop, then climb up the cliff on the left and push a boulder off the cliff onto the sentry to destroy it. It will drop ancient parts that you can collect.
After you destroy the sentry, go back down and push the boulders out of Yunobo's way, then have him follow you. Up ahead, there is a cannon that Yunobo will automatically climb into. Use the cannon to hit the Divine Beast.
After that, go forward on the path until you encounter more sentries. Make Yunobo stay still, then climb up the cliff and use Magnesis on a metallic crate to hit the first sentry. Put a crate on the next ledge, climb to it, and destroy the second sentry. Then go back to Yunobo and have him follow you. When you reach the sentry that isn't moving, use Magnesis on a metallic bridge nearby. Put it on the ledges so it blocks the beam of the third sentry.
The path ahead goes up a slope, and you will find a moblin. Kill it, then use the nearby cannon to hit the Divine Beast again.
Continue on the path, and you find two sentries, one circling the other. Throw a bomb at the updraft, then detonate it at the top. Keep throwing bombs at the updraft and detonating them until you have killed the sentries.
Go farther ahead, then make Yunobo stop before you reach the beam of the first sentry. Sneak past the first sentry, then float on the updraft to reach the top of the cliff. From there, you can throw bombs and detonate them to get rid of the three sentries in this area. Afterward, go back and have Yunobo follow you.
Go farther ahead and you find two Moblins. Kill them, then use the nearby cannon to hit the Divine Beast once more. It will retreat to the crater of the volcano, and you automatically glide down to it and register it as a travel gate. When you are ready, you can enter Divine Beast Vah Rudania.