Can anyone provide some spoiler-free clues as to where to start looking for the quests that start this?
Obtaining the Master Sword isn't a side quest, if I remember. I just stumbled on it. Once you find it, you'll need 13 full heart containers.
(This answer is short/terse to limit possible spoilers. I think I might have already said too much.)
It is located in forest, covered in fog, which is north of Hyrule castle. Enter the forest from the south-east and you will see various torches. Walk in the direction that the embers are blowing. At a certain point, you will need to light your own torch and keep stopping, to see if the embers have started blowing in a different direction.