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.
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. The volcano will erupt along the way, so be sure to hide under a rock ledge to avoid the falling rocks.
Keep going north to reach Goron City. There will be a cut scene before you get there.
In Goron City, go straight ahead, but stop before the bridge over the lava. Go to the right to find 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.
Abandoned North Mine
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. Then be sure to rotate the cannon to the right to shoot some cracked rocks down below. This will reveal an updraft. You can float on that updraft to reach a cannon on a pillar up above. 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.
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.