Hyrule Field
Head away from Kokiri Forest and Link will meet Kaepora Gaebora - the infamous owl. Talk with him and set forth towards the north where Hyrule Castle awaits. Be sure to enter the castle during the day time because Link is unable when it is night out. After entering - continue until you reach the Market.
In the center will be a fountain with a young girl named Malon near it. Talk with Malon and continue towards Hyrule Castle. Upon entering the field adjacent to the castle - Kaepora Gaebora will speak to Link once more. Now climb up the vines on the right and continue towards the small bridge.
Do not cross it - go down the ladder near the start. Exit through the door and continue down the path. Once the hill goes from grass to dirt - be sure to climb up. You will find a guard. Go around to the right of the guard and continue down the small dirt path. Go all the way across the field until you find some vines on a wall.
Climb up the vines and continue towards Hyrule Castle. Jump down behind the gate and hop into the moat. Swim around to the opposite corner and climb out. Go down the path towards the boxes to find Malon's father sleeping. Speak with him and then exit back through the gate. The guards will throw you out. Now go back to the vines to find Malon. Speak with her to get the Weird Egg.
Equip the Weird Egg and travel back to Malon's father. Wait for a day and night cycle to pass so that the egg hatches into a Cucco. Show the Cucco to Malon's father to wake him up. Talk with Talon - Malon's father - once he is awoken. Talon will run off to find Malon. Now push the 2 wooden crates into the water where Talon was sleeping.
Castle Courtyard
Climb onto them and jump across to the small entrance. Travel through it to reach the Castle Courtyard. Continue down the trail until Link comes across a guard. Make sure to sneak pass the guard - you can do so by waiting for him to get to the front and then quickly rolling around the back. Roughly the same thing can be done on the second set of guards.
Link can sneak past the third set of guards by climbing on top of the overpass and walking across the wooden beam. The next set of guards are slightly harder to sneak by. You must wait until one is on the opposite side and walk towards him some. Now stay between both guards - out of sight - and eventually Link will be on the other side.
Follow a similar pattern for the final set of guards. Travel through the doorway to find Princess Zelda. Speak with her for a cutscene. Be sure to tell Zelda that you have the Spiritual Stone of the Forest and continue with the cutscene. Zelda will give Link Zelda's Letter. Now travel back towards the area with the guards. Impa will now teach Link Zelda's Lullaby and transport you outside of the castle.
Saria's Song
Head to Kokiri Forest and travel up the ledges opposite of where you got the Kokiri Sword. Enter the large tree-like opening to find the Lost Woods. Now go right, left, and right, left, straight, left, and right to find the Sacred Forest Meadow. Kill the Wolfos near the entrance to unlock the path forward.
Going through the opening to enter the maze. Now travel to the right and continue down the path – killing the Mad Scrub – and continue down the path. Once you reach a set of stairs - go up them to find a couple more Mad Scrubs and eventually Saria, who will teach Link Saria's Song. Now exit back through the Lost Woods and go to Hyrule Field.
Part 1: Great Deku Tree | Part 3: Death Mountain