After your attempted rescue of Zelda in Hyrule Castle, you end up in the dark world of Lorule.
Get a Bottle
Use a bomb on the left side of the rear wall of the Vacant House (south of Lorule Castle) to open a hole. Go in and open the chest to get a bottle.
Buy Your Rented Items
After you end up in Lorule, you can go to where your house would be in Hyrule and find a warp fissure there. Use that to go back to Hyrule. Visit Ravio's shop. He now allows you to buy the things that you have rented. If you buy it, it's permanently yours. You will no longer lose it if you fall in battle. In addition, Ravio takes 50% off of the price of the first rental item that you buy. You might want to buy the Ice Rod or Fire Rod as your first item, because they are more expensive than everything else.
When you have the bombs, go to the area just north of the southeast lake. You can bomb the wall behind the sign that says that a strange creature lives inside. In there, you find the Mother Maiamai, who asks you to find her 100 missing children. They will make a sound when you are near them. She gives you the Maiamai Map to help you find them. When you go outside, you find one sitting right there, but you can't pull it off of the wall and none of your items work. The only way to get the Maiamai is to merge with the wall, walk behind the Maiamai, and pop out to make the Maiamai pop off of the wall. Not all Maiamai will be stuck to walls. Some are hidden under rocks or even under water.
Every time you bring ten Maiamai back to the Mother Maiamai, she will upgrade one of your items, but only if you have bought it from Ravio. She can't upgrade rental items.
Rescue the Sages
You can do the dungeons in almost any order. Here is a guide that explains how to reach all seven of the dungeons.
There are no prerequisites or required items to complete this dungeon. This dungeon is in Thieves' Town, which is where Kakariko Village is in Hyrule. You can just walk there from where you start in Lorule. The door of the dungeon is in the north middle part of town. Check on the door under the leftmost statue, then say the correct song lyrics to open the door. See the Thieves' Hideout page for more details.
The Desert Palace is actually in Hyrule. You need the Sand Rod in order to get in and complete the dungeon. The Sand Rod is obtained after completing the Thieves' Hideout dungeon.
To get there, you first need to go to southwest Lorule to find a warp fissure west of the Swamp Palace. Use the warp fissure, then walk west into the desert of Hyrule. In the desert, you have to use the Sand Rod to reach the raised southwest area, and use the Sand Rod again to reach the northwest path that leads to the Desert Palace.
To complete the Swamp Palace, you need the Hookshot and Tornado Rod. The Swamp Palace is south of the Vacant House in Lorule. Before you can complete the dungeon, you have to bring a Bomb Flower and blow up the cracked rock just south of the Swamp Palace entrance.
The Bomb Flower guy lives in a house south of Thieves' Town. Ask to borrow the Bomb Flower and pay the 200 rupees. Go through the east exit of his house and check on the bomb flower. It will start following you. Walk past the cracked rock with it and then hit it to activate it. The bomb will explode, breaking the cracked rock. Now you can take a bomb flower whenever you wish.
Keep in mind that if an enemy hits the bomb flower, it will activate. Take a bomb flower to the Swamp Palace and put it in front of the cracked rock to blow it up. Now you can enter the Swamp Palace.
The Turtle Rock dungeon is in southeast Lorule, in the lake. You will need the Ice Rod in order to complete the dungeon. To get into the dungeon, you have to rescue three turtles. See the Turtle Rock page for more details.
The Skull Woods are in northwest Lorule. To get there, go to Rosso's House in Hyrule and go inside. Use the warp fissure in his house. Then in Lorule, merge with the wall to get past the bars, then go south until you can't go any farther south. From there, go west and then north. You will be in the Skull Woods. There is an entrance to the woods up the path.
The Dark Palace is in eastern Lorule, where the Eastern Palace is in Hyrule. To complete it, you will need either the Hammer or the Bombs, or both.
The Ice Ruins are in northeast Lorule, on the mountain. To complete it, you need the Fire Rod and the Hookshot. To get there, first warp to the Tower of Hera, then go east and use the hookshot to cross the bridge. Go east and enter Rosso's Ore Mine.
Use the moving platforms to reach a ledge where two Lynels are walking around. Merge with the wall on the right and go to the other side, and get on the moving platform when it comes over to you. The next series of platforms are much more difficult. You have to fall onto the platforms below you. After you do it all correctly, you reach a door to the outside. Go out. Go west from the door to find a warp fissure to Lorule. Go through.
Once in Lorule, save at the weather vane, then go east and go into the cave. Use the moving platforms to reach an area with some ice statues. Go west to find a big ice statue. You can kill it with the sword or the fire rod. Afterward, wait for a moving platform to appear to the north. Step onto it and use the platforms to make your way to the exit.
Outside, use the weather vane. Then use the Fire Rod to get rid of the statue blocking the entrance of the Ice Ruins.