Lon Lon Ranch
Head towards the center of Hyrule Field to find Lon Lon Ranch. Travel towards the end of the trail to find a horse corral. You will find Malon standing next to a horse named Epona. Talk with Malon 3 times and then take out the Ocarina. Malon will now teach Link Epona's Song, which be really helpfully later in the game.
Now it is time to head towards Kakariko Village. Exit from Lon Long Ranch and go back towards Hyrule Castle. Continue down the path to the east towards Death Mountain. Travel across the small bridge and go up the stairs to find the entrance for Kakariko Village.
Kakariko Village
Getting the Sun's Song
Head to the east of Kakariko Village and enter the Graveyard. Go all the way to the back of the Graveyard to find a large headstone with the Triforce in front of it. Now play Zelda's Lullaby while standing on the Triforce to open up an entrance in the tomb.
Drop down into the Royal Family's Tomb and continue down the path. Kill all of the Keese to open the door ahead. Enter the door and continue down the path past the ReDeads. At the end of the tomb will be some writing behind two torches. Read the writing to learn the Sun's Song.
Getting the Hylian Shield
Exit back to the Graveyard and play the Sun's Song until it is night time. Now pull the headstone with the flowers - the second one from the left - on the first row of graves to reveal a small hole in the ground. Jump down into and open the small chest for the Hylian Shield. Now exit the graveyard.
Go up the stairs to the north of Kakariko Village to find the entrance to Death Mountain, which is blocked by a locked gate. Show the guard Zelda's Letter to be allowed through the gate and onto Death Mountain. Go through the gate to reach the Death Mountain Trail.
Death Mountain Trail
Climb up the trail and Link will find a cave blocked by a large amount of rocks beside a Goron. Now continue going up the trail and the player will find a split in the path - one heading higher up the mountain and one heading towards Goron City. Take the path to Goron City and go inside the entrance.
Goron City
Once inside Goron City - jump all the way down to the bottom floor. Go to the locked door and play Zelda's Lullaby. Now enter the next room and play Saria's Song for Darunia - chief of the Goron's. After speaking with Darunia, Link will receive the Goron's Bracelet. Exit the city and head towards the right to find a Goron sitting beside a Bomb Flower.
Pick up the Bomb Flower and toss it over the ledge in order to break the rock blocking up the cave from earlier. Now travel back down Death Mount Trail until you reach the cave. Go inside and get ready for another dungeon - Dodongo's Cavern. Link must use a Bomb Flower once inside to actually unlock the dungeon.
Part 2: Hyrule Castle | Part 4: Dodongo's Cavern