The Goron Sidequest is how to get the Mirror Shield. I put instructions for this quest in the main walkthrough, but here it is separately for those who just want this information and not the rest of the walkthrough.
To complete this sidequest, you will need the Mole Mitts, the Grip Ring, and the Roc's Cape.
To start off the Goron Sidequest, you need to use the wind vortices in the Lon Lon Ranch area to reach the Goron in the southwest corner of the area. Fuse Kinstones with this Goron, and the result is that the Goron Merchant will come to Hyrule Town. This opens up the next part of the quest.
The Goron you fused Kinstones with will try to dig open the cave at the southwest corner of Lon Lon Ranch but seems unable to do it by himself. To fix this, go to Eenie and Meenie (the farmers in the Eastern Hills) and fuse Kinstones with the farmer who is nearer to the top of the screen (that's Eenie). This Kinstone Fusion gives the Goron the power to break down the rocks blocking the opening to the Lon Lon Ranch cave. Unfortunately, there are more walls inside, all of which are too big for that Goron to break down by himself.
Once you get the Mole Mitts, you will discover some caves that you can explore after breaking through some soft wall barriers with the Mitts. In some of these caves, there are strange spots known as Mysterious Walls. You can't talk to Mysterious Walls, but you can fuse Kinstones with them (so I guess a Mysterious wall is really just a Kinstone Piece embedded in the wall, with an empty spot next to it). At any rate, when you fuse Kinstones with a Mysterious Wall, it makes a new Goron appear in the Goron Cave in Lon Lon Ranch.
Where are these Mysterious Walls, you ask? Here's the list.
Once you have fused with all five Mysterious Walls, the Gorons will successfully break through the walls in the Goron Cave. You can reach some rupees and an empty bottle in a chest. Talk to the Gorons all the way in the back, and you can fuse Kinstones with one of them. This final Kinstone Fusion will awaken Biggoron on Veil Falls (whom you might have seen sleeping when you visited there before.)
All you have to do now is give Biggoron your shield, wait around a while, and then go visit Biggoron again to get the Mirror Shield.