Child Link
Piece of Heart #1
Grab the Cucco near the iron gate at the start and carry it until the middle section, near the Spider enemies. Use the Cucco to jump and glide to a hidden ledge. Then, climb the vines to the elevated area and jump over to the southern platform.
Drop down into the Grotto, grab a new Cucco, and carry it to the elevated area. Use it to jump and glide to the isolated platform on the northern side, where you'll see the Piece of Heart.
Piece of Heart #2
Grab the Cucco near the iron gate at the start and carry it until the end of Zora River. Just before the exit, you'll see the Piece of Heart on an isolated ledge to your left (in the northern part of the map). Use the Cucco to glide and grab it.
Child and Adult Link
Piece of Heart #3
As Child Link, plant Magic Beans in the soft soil next to the Death Mountain Crater teleport location.
Then, as Adult Link, return to the same location and ride the Magic Plant to reach the Piece of Heart.
Piece of Heart #4
As Adult Link, learn the Song of Storms.
Then, as Child Link, play the Song of Storms at the Frog Tune minigame near the middle section of Zora River. The frogs will reward you with the Piece of Heart.
Piece of Heart #5
Requires learning Zelda's Lullaby, Saria's Song, Song of Epona, Song of Sun, and the Song of Time.
As Child Link, perform all these tunes to the Frogs (might also require playing the Song of Storms and obtaining Piece of Heart #4 beforehand).
Then, play the frog jumping minigame, where you'll have to play the following notes in order:
L Y A R Y A R L R L R A Y L