Climb the ladder in the entrance room. You will be on a ledge. Turn right and start to run - you will be climbing uphill (the character will automatically jump over the small gaps if you keep constant pressure on the joystick). Run until you come to a treasure chest, and open it (from now on, open every treasure chest you see). Then continue running until you come to a door. Enter.
In the next room, use the Deku Shield to bounce the enemy's rocks back at it to defeat it (from now on, always defeat every enemy in a room). The door will open - go to the next room. The floating ledge will collapse when you jump on it - don't worry - you won't die. Find the two treasure chests in this room - one will have the slingshot in it. Use that to shoot down the ladder on top of the exit, and climbed the ladder to exit the room.
Walk back to the part of the ledge where you got the first treasure chest. Shoot all three spiders on the wall with the slingshot, and then climb up on the vines. Climb up and to the right, and then jump off onto the higher ledge. Walk around until you see a door and enter it. In this room, you will see one lit torch and one unlit torch - always be suspicious when you see something like this. Use a Deku Stick to transfer the fire from one torch to the other - this will open up the bars that were blocking the exit from the room - don't exit yet, though. Now step on the switch and jump on the "islands" to get to the other parts of the room and to get the treasure chests. Always kill spiders and collect the gold tokens that they leave behind - they are important.
Now leave the room and find a spider that descends from the ceiling. Kill the spider and then walk slowly up to the brink of the ledge. You will notice that the camera switches to the overhead view. Try to line up the center of the floor (which the web is blocking) and jump off aiming for the center of this web. If you land correctly, you will break the web and fall to the lower level. If you don't break the web, you will have to climb back up to the top and try again. Step on the switch in this room to light a torch. Also, open every treasure chest in sight. Light up a Deku Stick in the torch, cross the shallow part of the water to the place where the door is blocked by a spider web, and run into the web carrying the fiery torch. The web should burn and then enter the door.
Defeat the enemy in the next room and shoot the gray stone above the door with your slingshot to open the door. In the next room, dive into the water in the left part of the center of the lake and hit the switch. Jump onto the floating raft and go to the other side, push the block underneath the high exit, and climb up the block and go out the door. In the next room, light the 2 unlit torches with a Deku Stick, and go to the next room. Here, light a Deku Stick and burn the web on the right. Then proceed to crawl through the small opening into a familiar-looking room, except you'll be on the other side of it. Push the block into the water to form a type of "bridge." Cross this "bridge," light a Deku Stick, cross back and stand above the hole with the web covering it. Swing the fiery Deku Stick, and you should burn the web.
You will fall down to the lowest level. You will now face 3 enemies. Knock them out of their holes in this order - middle, right and left. After they are out of their holes, kill them all with your sword. Now, enter the door to face the leader - Gohma. Gohma is the one enemy where I think that the Deku Nuts actually serve a purpose. Stun Gohma with the nuts, and then strike the eye with your sword a few times. Gohma will retreat, climb the wall, drop enemies from the ceiling, which you will have to kill, and then Gohma will drop from the ceiling again. Attack the boss in the same manner as before, and it will retreat again. Repeat this until the boss is defeated. Then, pick up the heart container that the boss leaves behind when it dies (don't ever forget to do this - sometimes the heart containers are easy to overlook). This will add another heart to your maximum health level. Then, exit the level via the green light in the center of the room.
The Deku Tree will then go into a long story about life in general, and will give you a stone, and will then die. He will tell you to leave Kokiri Forest and to go to Hyrule Castle, which is covered in the next section.
Proceed to Hyrule Castle...