Ascend to the Second and Third Floor
Walk into the first room. Turn to the left, there should be a ladder (do not use the vines, there's a Skultulla crawling on them). Jump across the log (be wary of the Keese) and in front of you there should be a large chest and a Gohma egg. dispatch the egg before it hatches (or kill the larvae with two sword strikes). There is also another egg up above you that will drop down if you run underneath it. Open the chest to obtain the Dungeon Map.
Adjacent to the chest is an oddly placed crate, roll into it and you will find your first Gold Skulltula. Since the only door on this floor is covered in a spider web, the only way to go is up. Climb the vines to the third floor.
Explore the Third and Second Floor
Find the golden torch and dispatch of the lone Keese to cause you less pain. Look around for a switch not far from the torch, and step on it. It will light the torch, burning the web covering it. Go through the door.
In this room, if you light a Deku Stick and light the torch in the center of the room, a chest will fall. This chest will hold a Deku Shield if you don't still have one, or 5 rupees if you do. What you have to do in this room is kill all the enemies. There is an egg on the top of the room that you have to make fall to be able to do this. All the rest are pretty easy to see. You can press the switch on the same platform as the door you came in to make three platforms rise out of the ground to get to the higher ledges. Once you kill all the enemies a chest will appear. Open it to take the slingshot, and leave the room.
Kill the Keese as soon as you get out of there to avoid losing your shield if it lights itself on fire and touches you. Then run back to the vines that you climbed to get up here. Break the egg hanging above the vines with you new slingshot, and climb the vines back down (if you jump towards the vines Link should grab them). Kill the Gohma Larva if need be. Then, light a Deku Stick and take it to the web covering the door from earlier, and go through it.
In this room, break the eggs, and then kill the Giant Deku Baba (targeting him and shooting the slingshot at hims seems to work best) and then go through the door to the next room. You will see a floating platform in this room. DO NOT STEP ON IT! A spider will come down on top of you while the platform begins falling causing damage to you. Instead, shoot the egg in front of the treasure chest, and break another one on the ground. Shoot down one of the spiders climbing on the vines, then climb the vines. Run behind the chest and shoot the backside of the spider hanging from the ceiling to kill it. Then you can open the chest and get the Compass. You can now try to use the floating platform to get back to the door you came in from, or you can shoot the ladder down with you slingshot and climb it. Leave the room, and get back to the central room of the dungeon.
Jump down to the first floor of the dungeon and use the torch to burn away the web on the ground (rolling on the web seems to work best for lighting it on fire) and drop down into the basement.
Through the basement
Climb out of the water and onto the platform that has two Deku Babas and a switch on it. The switch lights a torch on the platform across the room. Use this torch to burn away a web with a chest behind it. This chest contains a Deku Shield if you need one.