Piece of Heart #1 - Hyrule Castle
You'll find this by the castle's eastern wall. Lift the small rocks to reach it.
Piece of Heart #2 - Sewers
You'll probably spot this as you travel through the Sewers underneath the Sanctuary early in the game. Return with the merge ability, then cross through the wall from the walkway above to reach it.
Piece of Heart #3 - Blacksmith's House
This is on an upper ledge behind the Blacksmith's house. To get there, merge into the front side of the house. Slip back here to grab your prize.
Piece of Heart #4 - Blacksmith's House
This is found inside a cave underneath the Blacksmith's house. When you obtain the Titan's Mitt, you can lift the boulder blocking the cave in front of the house.
Piece of Heart #5 - Kakariko Village
This is found inside the well. At the northwest corner of the village are two entrances to said well. Lift the rocks on the upper ledge, then hop in from the cliff. Head up the stairs inside to grab it.
Piece of Heart #6 - Kakariko Village
When you reach Thieves' Town in Lorule, look for a rift in the middle of a wrecked house. Use this to return to Hyrule…right inside of this lady's locked home! She'll give you a Piece of Heart and leave her door unlocked from now on.
Piece of Heart #7 - Cucco Ranch
The ranch is located just south of Kakariko Village. Speak to the Cucco Girl to play her game; you must dodge flying cuccos for 30 seconds. Complete all three levels to earn a Piece of Heart.
Piece of Heart #8 - Rupee Rush
Rupee Rush is located just west of the Cuccoo Ranch. The goal here is to grab as many Rupees as you can within 30 seconds; the thing is, there is no timer on screen while your running around. You'll have to get 100 Rupees during the game to earn a Piece of Heart. It sounds tough, but you receive bonus Rupees the close you get to the game's ending timer.
Piece of Heart #9 - South Hyrule
This is found in the area south of Link's House (it's equivalent to the Swamp Palace in Lorule). Among the columns is a Piece of Heart blocked by several columns. Directly southeast of here are two heads against a cliff face. Place a Bomb between them to reveal a hidden cave entrance. Follow the cave as it wraps counterclockwise to emerge at the Piece of Heart.
Piece of Heart #10 - Zora's Domain
This is found along the river that leads to the Zoras' waterfall home. When you drop over a waterfall, look for the cave entrance behind the falls. Enter to find a Piece of Heart inside.
Piece of Heart #11 - Rosso's Ore Mine
Head to the northeast section Death Mountain. The fire-breathing enemies here are guarding the entrance to the mine. Inside, you'll follow some platforms until you reach a central one guarded by two more fire-breathers. You have a few platforms to choose from here; take the left path (you'll need a Hammer to launch from the frowning faces) to eventually reach a Piece of Heart.
Piece of Heart #12 - Eastern Palace
Directly south of the Eastern Palace entrance is a cracked wall behind an Armos. Bomb the wall, then enter the cave. Merge to cross the gap left; there's a Piece of Heart on the ledge.
Piece of Heart #13 - Eastern Palace
Head farther south from the Eastern Palace to find a series of pegs surrounding a Piece of Heart. Pound them out of the way with the Hammer to grab it.
Piece of Heart #14 - Lake Hylia
Head to the enclosed area northeast of the lake proper. Here you'll find the Racing Bro, who runs Hyrule Hotfoot. Complete the second race to earn this. Yes, it's almost unfairly difficult to clear, but you'll beat it with persistence.
Now, you're probably wondering just how you're supposed to complete these footraces. Without the Pegasus Boots, you just won’t be able to finish in time. To get them, you need to talk to the Shady Guy in Kakariko Village. Speak to him by merging into the cliff nearby and sneaking up behind him.
Piece of Heart #15 - Death Mountain
Just before reaching the Tower of Hera, you'll merge across this gap. Drop to the central cave and head through. Instead of entering a proper cave, you'll walk beneath the volcano and discover a Piece of Heart.
Piece of Heart #16 - Lost Woods
This is found right in the center of the forest. In the central log is a small crack in the rocks. Merge through here to find a Piece of Heart.