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Now you're inside the Deku Tree. Link is facing a large round chamber (the main room of the Tree) with a giant spiderweb covering a pit in the middle of the floor and a raised staircase leading upward around the rim. To climb that staircase, go inside the rim and climb the ladder leading up, Around on the other side, keep bypassing the vines leading farther up. Get the Map from the treasure chest. It really isn't too much good now because you don't have the Compass yet.
Keep going until you reach a small alcove with a closed door and enter after Navi tells Link how to open doors. Really, you'd think he'd know something like that already. In this room is a Deku Scrub. Z/L-target it, and reflect the seed it spits at you with your shield. I suggest experimenting with targeting to learn how to center the camera behind you, target enemies, switch between them, etc. Into the next room, jump across the gap and open the treasure chest for the Faerie Slingshot.
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Breaking the Web
You may have noticed as you were coming up that there were spiders called Skulltulas hiding along the vines near the map. You have to climb these, so go back down, get out the Slingshot and use them for target practice. You might want to take this opportunity to learn the center of sight. This is how you can have Link be able to hit any target without having to use L-targeting. For the Slingshot, this is at the point parallel to the halfway point of the right fork of the Slingshot when it's cocked and ready to fire.
Climb up the vines, go up the staircase and enter the room. You'll see a depression spanning the room lengthwise, and on the other side is the alcove with the Compass. You can get it if you want, but I don't think you'll need it. There is something else here that you'll likely want. Go to the small alcove on the left side along the wall and kill the Skulltula, grabbing the token. This is a Gold Skulltula, and there are 100 of them in the game. They'll be useful to Link later on - most if not all Zelda games have something like this. The real objective of this room is to get in and out as quickly as possible, and it can be tricky. See that torch on the right side? Get out a Deku Stick, light it, run over to the unlit torch and light it (just walk up and touch it). Then, put the Deku Stick away to reset the invisible burn timer on it, re-light it and go out. You have to do this quick, or the stick will burn out.
Now, go to the left as you exit the room. There'll be an opening in the cobweb wall. QUICKLY throw a Deku Nut to stun the Skulltula that will drop down (before it hits, so it doesn't block your path) and jump out of the opening, landing in the cobweb below with the still-lit Deku Stick. This allows you to burn through and make it below ground.
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Lucky for Link there's water at the bottom of the pit below the cobwebs. Swim out and climb to land, killing any enemies you find. There's another Gold Skulltula on the vines leading to the surface, so get it if you want - there's not rush, you can come back to the Deku Tree anytime you want. To the right of where Link landed, there'll be a switch next to a golden torch and some web. Step on it to light the torch, then kill the Skulltula on the wooden bars in the water.
Go back across to the left after lighting a Deku Stick. Burn through the web and go to the next room. There's a spiked log and a moving platform. Jump into the water, press the switch, go back up and jump on the platform to pass under the log and get to the other side. In the next room, kill the Deku Baba here, light the two unlit torches and go into the door that opens. In the next room, don't rush in - a giant Skulltula is waiting just above the entrance to attack Link. Target it, make it drop down, then shoot its underbelly and attack it with the sword. Two normal strikes or one jump attack (A whilst targeted, or just holding down the L button). You'll see two cobwebs - there's a Gold Skulltula behind the one on the diagonal left, but you can't get it yet, so go right. Burn through the web and enter the crawl space.
You'll notice you're back where you started here, but on the higher level. Push the block into the water, jump across, light a Deku stick and come back through with it and burn the web in the ground.
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Even Further Underground...
After landing in the highly-convenient water (seriously, Link's got to be one of the luckiest video game characters there are, with all the watery landings he seems to be getting) climb onto land. There are three Deku Scrubs ready to attack, and it has to be done in the right order: middle, right then left. Any other order won't work. After Link kills them all, the door to the boss will open. I'd advise having plenty of Deku Seeds for the Slingshot, a full Heart Container set (only three hearts, but still) and a basic understanding of how targeting works. This first dungeon in Ocarina of Time was mainly intended to let you learn the basics of moving and fighting - don't think they'll all be as easy as you likely found this one. Anyway, Link's about to be up against....
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Parasitic Armored Arachnid: GOHMA
...hmm. Ominous. The walls are all green and glowy, and the door just slammed shut behind Link. There's a rustling sound up on the ceiling. Go into first-person view (C-up) and look toward the red dot moving along the ceiling.
Well, lucky Link, you just caught the attention of a giant spider who wants to drop down and eat you. I assume that's what's been spinning all the webs you've been burning through. It doesn't look too happy, and that's a bad sign for something that just presumably rotated its eye vertically 360 degrees. So, get out your Faerie Slingshot, target and keep your distance, just out of its striking range. When the eye turns red, let fly with a Deku Seed. This will stun Gohma. Now, move in and attack with the sword. This is a pretty standard strategy in Zelda games: weaken the boss with whatever items you have on hand, then move in and slice and dice with the sword.
Eventually, Gohma will recover and retreat to the ceiling to drop eggs. Kill the larvae, or if you feel like finishing the fight quickly, try to shoot her eye again. She'll fall down where you can continue your attack. 5 Jump Attacks should do the trick all told, or 10 normal attacks. After Gohma melts and vanishes into black crumbliness, grab the Large Heart Container, upping your life to 4 Hearts, and step into the blue pillar of light to be transported out. Congratulations, you've just beaten the first dungeon in Ocarina of Time.
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Definitely Not Arbor Day
Well. The Deku Tree has more information to tell Link, so settle in. He tells you that the person who cast the curse on him was from the desert, and he was ultimately after the entrance to the Sacred Realm, where the Triforce of the Goddesses can be found. Three gods created the world of Hyrule: Din, Farore and Nayru. Power, Courage and Wisdom, and each had a certain responsibility. Din made the earth, Nayru made the spirit of law (basically the sky and water and such) and Farore gave the planet life. After they were done, they converged in the sky, leaving the Triforce, three golden triangles, as the mark of their presence.
Unfortunately, Link's work was for nothing. The Deku Tree was already dying. After telling Link to never allow The Man In Black (as the Deku Tree calls him, emphasis mine) to get the Triforce, he bequeaths the Spiritual Stone of the Forest to Link. He tells Link to go to Hyrule Castle, then dies and turns to petrified wood. There's nothing for you here in Kokiri Forest, so head out and through the west exit, where Saria is waiting. After a dialogue, she gives Link the Faerie Ocarina as a reminder of her. Now all of Hyrule is just waiting to be explored.