This is the first dungeon of the game, and the first room is a pretty large one, with a web covering the middle of the floor. When you get control of Link, go around and kill the Deku Babas, which will take two regular sword hits each. After you kill them, go up to the different-colored wall, which will prompt Navi to mention that you can climb it.
Climb up the vines and continue along the path to a chest, which contains the Dungeon Map. All of the dungeons have one of these, which will certainly help you when getting lost. You can't climb the vines here yet, as the Skullwalltulas will knock you off, so continue in the same direction and hop over the gap, to the door.
Go through the door and iron bars will come down, locking you in. There will be a new enemy in this room, the Deku Scrub. It will sit in the middle of the room and spit Deku Nuts at you, which you bounce back at it with your shield. Bounce a seed back at it, which will cause it to come out of the ground and run around; if you close in after bouncing the seed, it won't have time to run around.
Once it stops, it will talk to you and tell you a tip, then it will go away and the doors will open. Go through the other door, where you'll be in a small room with a suspended platform in the middle. Hop onto the platform, which will cause it to start shaking and fall to the ground. Your goal is to hop to the other side as quickly as possible; if you fall, you can use the vines to climb up to the platform. Open the treasure chest to receive the Fairy Slingshot.