To complete the Song of the Hero, you need to go to Eldin Volcano and learn the Fire Dragon's Song.
Golo Game-Breaking Glitch
The Fire Dragon
When you go down to Eldin Volcano, the volcano will erupt and you will get blown away on the Sailcloth. You will end up with only two hearts, and will wake up in a prison cell of the Bokoblins. All of your equipment has been stolen.
Recovering your Stolen Equipment
Walk around your jail cell, pick up and throw pots for hearts, and eventually Plats will appear. He will give you back your Mogma Mitts.
Dig into the hole in the dungeon room to burrow out to a more open area. Look for the little hut to the north where there aren't any enemies in the way. Sit on the stool for a while to regain all of your hearts.
Go through the eastern tunnel to reach the outside. You can save at the Bird Statue if you wish. In the Wii and Wii U versions of the game, there is a Gossip Stone nearby that can show you hint movies.
Follow the path northwest and drop down from the ledge to meet Plats again. He will warn you not to let the bad guys see you, and will mark your map with the locations of all of your remaining items.
Climb up the wall behind where Plats appeared. Go a bit to the east and you will see a Bokoblin walking back and forth. Wait for it to walk to the east end of the path, then go along its path and drop down to the north. Go east, but be careful of the Red Chuchu that appears, and go east to the soft soil. Dig in it. Go around and break rocks with your mitts and hit the bombs to reach the other light spot where you can emerge.
Go east along the path and run up the ledge. You will see a brief cutscene near the lava that shows you what lies ahead. You have to carefully drop down near the searchlight tower and dig into a burrow hole. There is a Moldorm in this burrow hole, so be careful. Go to the upper-left spot of light and emerge there, then go up the ledges to reach the chest that contains the Gust Bellows.
Use the Gust Bellows to blow away the nearby chunk of lava across from the chest that you just opened, then climb up the east ledge, and run up the southern ledges. Approach the area to the south where two Bokoblins are walking around, and there will be another brief cutscene showing the path ahead. Carefully go past the two Bokoblins that are walking around. Even if one of them sees you for a moment, you can hide behind a wall and it will stop looking for you after a few moments. Make your way to the soft soil to the southwest, and dig down into it. Get past the Moldorm and emerge at the upper-right spot of light.
Drop down and get the Clawshots from the chest. Use them on the Peahats to the north, and then on the vines at the very end.
Drop down onto the ledge at the base of the leftmost vines, then walk around the ledge. Platforms are floating from north to south here. You can get into the cave entrance to the west by jumping across the rocks that float past, but the cave just contains healing items and a Gossip Stone that will appear near the butterflies if you strum the harp.
Jump onto one of the floating platforms, and ride it to the first ledge with a torch on it, on the right. Go up and you will see a cutscene of a Bokoblin walking around and a search light. Make sure that the Bokoblin is not above the ledge, and climb up, then go around behind the Bokoblin until you can use the Clawshots at the north side to get onto the vines to the east. Pick up a Bomb Flower and use it to knock down the wooden tower to the south. Then watch to make sure the Bokoblin won't see you, and open the chest past where the tower was to get your whip back.
Go north to the bridge that has bars on the end. Use the whip to activate the Peahat just north of the bars, then use the Clawshots on the Peahat to go to the other side of the bars. You can save at the Bird Statue if you wish. Go southeast to the Bomb Flowers at the end of the path. Use them to destroy the rock walls in the east tunnel. Then run forward into the large air vent to reach the top of the eastern slope.
You will see Plats up ahead. He will say that there are bad guys gathering in the cave up ahead who seem to have found a great treasure. You can't go that way yet.
Go north across the bridge and use the Clawshots to kill the Keese. At the end of the tunnel, a cutscene will show you a Peahat and some floating lava platforms. If there is a Goddess Cube here, you can't activate it yet because you still don't have your sword. For now, go to the floating platforms. You can use one to jump across and reach the cave, which just contains healing items and another Gossip Stone. There is also a rock at the top of the ledge that you can't get rid of until you get bombs back.
Back outside, use the Clawshots on the Peahats to go south, toward where the Slingshot is. When you're hanging from the last Peahat, drop down when a floating platform is under you. Go to the west side with the Bomb Flowers and pick one up when another floating platform comes close. Jump onto the floating platform and throw a Bomb Flower under the wooden tower, then wait until you can jump off on the southern side of the ledge with the tower. When the tower is gone, you can go to the blue chest and get the Slingshot. Use your Clawshots on the Peahats to go back north to the last area.
Go north up the slope and go through the hut area. You will find some Bokoblins walking around. You have to shoot them with the Slingshot to stun them, then run past them, but beware of Red Chuchus that can appear in this area as well. If a Bokoblin sees you, you will only be sent a short way back to the start of the hut area. When you reach the wooden tower, use the Slingshot to stun the Bokoblin on top of the tower and burrow into the hole near the base of the tower. Go to the northern spot of light and emerge there, then open the chest here to get your bomb bag back. Dig at the dirt here to reveal an air vent, which sends you to a ledge with Bomb Flowers that you can use to restock your bombs if needed. Use a bomb to knock down the wooden tower so you can safely go past.
You can use bombs to kill the Bokoblins and Chuchus to the east, or just use the Slingshot to stun them. If you were wondering what was beyond the bombable rock in the cave near the Peahat to the east, it is just some bombs and pots, which you can go gather if you wish. Otherwise, go to the northeast area where you can go north across the bridge formed by the wooden tower. Go through the cave with the Fire Keese, cross the bridge, and go southeast. Go past Plats, and in the hot room, use bombs to break the rocks, then go up the east ramp and go into the cave.
You will find your sword at the top of a ledge very close by. Pick it up by following the on screen instructions. Go north and you can see some Bokoblins guarding a chest. You can save at the Bird Statue if you wish. There is a fairy in the soft soil next to the statue.
Go up to the area to the east that Fi mentioned. Kill the Cursed Spumes with bombs, then ride on a floating platform. Use a Skyward Strike on the Goddess Cube, then go up the stairs, but don't go through the door at the top yet. Instead, go to the ledge north of that door, and go through the cave there. Stab a Water Fruit, then throw it at the lava up ahead to go across, but be sure to throw it far enough forward that you will be able to jump across the far gap. Now that you have your sword, there is no more need for stealth, so just drop down and kill the Bokoblins and Fire Keese, then open the chest to get the rest of your stuff back. Refill your Bomb Bag with the nearby bombs. Be sure to re-equip your shield.
Run up the southwest ledges, bomb the rocks at the top, then slide down the short slope below the rocks that you blew up, and go to the east, then jump on a floating platform to reach the stairs and go into the cave at the top.
Cut the rope gear with your sword, then use your Beetle to cut the two rope gears next to the gate. Go through and the Fire Dragon will teach you its part of the Song of the Hero. After this, the volcano will go back to normal.
Learn the Other Songs
If you haven't learned the other dragons' songs, you can go to the yellow column and visit Lanayru Desert to learn the Thunder Dragon's Song, and go to the green column and visit Faron Woods to learn the Water Dragon's Song.
If you have learned all three, visit Levias in the Thunderhead to learn the Song of the Hero.