Start by leaving Kokiri Forest. Saria will give you her Fairy Ocarina. Talk to the Owl on your way out and then follow the path to the castle.
Chances are you won’t make it and you’ll have to wait until morning to get in. If you stay on the path you won’t be hassled by the skeletons. Once the drawbridge is back down head inside the castle.
Once inside the market talk to Malon. She’s the red headed girl in front of the fountain. Now head out the other side of the market, towards the castle. There are tons of shops and things to do inside the market, so have a look around.
After you talk to the owl, he will fly away, and the screen will angle to towards some vines on the wall. Use them to get onto the gate.
First off though, go back to the market, then back to the castle. Malon should be there now, she's by the vines singing. Talk to her and get the egg. After a day it should hatch (in the morning) into a Cucco.
Climb the vines behind Malon and head towards the gate. Jump off the other side of the gate onto the trail (or go down the ladder) and follow the trail. When you see the two guards up ahead on the trail take a left across the field. Not to far in though, as there’s another guard near the gate. Once you get between them you’ll see another trail. Follow it (a small beaten path) but when it starts to point back to the main trail keep going left. Keep going west and you will come across a stone wall. Climb it to get up the ledge, and jump into the moat. Swim your way down the moat until you get to where the bars block the way, and in that corner you’ll find an elevated area that you can use to jump out of the moat. Then walk down to where you can see the man snoring. That’s Talon, Malon’s Dad. Use the Cucco to wake him up.
After Talon runs off move the Milk Boxes off the ledge and into the water. Use each one to make a bridge across into the castle.
Make sure its daytime before you enter. Now you’ve got to complete this maze quest:
Just go the opposite way as the guard on the first one.
Make sure both guards are on the opposite side before you run.
Take the upper platform across.
Follow the guards as they walk through the last one, and just make sure not to follow to closely.
The last room has Zelda in it, and this is where you continue on through the story. You can obtain twenty Rupees from hitting the window with your Slingshot. When you talk to her, of course you have the stone! Don’t lie to Zelda! One of the other important things Zelda teaches you is Zelda's Lullaby, which is something you be making quite a lot of use of in the future. She'll also give you the Note from Princess Zelda, which will allow you to continue through the game.
You’re now into the next part of the game. Time to go the Lost Woods.