With the hookshot, you can now visit almost all of the Dark World. Go north from the swamp to where the bomb shop is, then go east, hammer the posts and go north to the rock field, then north once more to where there is a broken bridge.
Just above the boulder, there is an arrow of grass pointing left. Stand at the gap (don't jump off the edge) and use your hookshot. You will latch onto the skulls across the way and be pulled across.
Go north, then west to where the graveyard is in the Light World. Dash into the pile of rocks just north of where you can get in, then walk behind a totem-like statue above. You can cut the bush and walk into the area that appeared to be blocked by those black skulls. (The pile of rocks in here hides 20 rupees.) Cut the other bush above to get to the ladder up to the ledge. Climb up, then stand at the left side of the ledge and use the Magic Mirror to go to the Light World. This ledge has a door in it, so go inside. Get past the pots and bomb the wall to reach a piece of heart!
Go back outside and use the warp you created to get back to the Dark World. Go down the ladder, and get inside the area that looks blocked by the black skulls. Stand in the top-middle of the enclosure and use the Magic Mirror. You are now inside an enclosure in the Light World that you normally wouldn't be able to reach because of the black stones. Watch out for the floating Poes, and stand in front of the grave and do a dash attack to move it back. Go down the stairs you reveal and walk north through this narrow chamber to find a treasure chest containing the Magic Cape! This cape renders you invisible and invulnerable, but it uses magic, so it won't work forever.
Go back outside and avoid the Poes again to get to the warp point back to the Dark World. From the "graveyard", head west to where the Sanctuary is in the Light World. There is a cave in its place; don't bother going in, the creature inside is just selling advice that you don't need. Go west from here, then north. Get to where the cave is in the Light World that leads to Death Mountain. Pull up the boulder and go into the cave. Kill the hardhat beetles and make your way upstairs. Use the hookshot to get beyond the hole. Your way appears to be blocked by a blue bouncy thing, but now that you have the Magic Cape, you can turn invisible and walk right through it! Do that (taking off the cape when you're through so you don't run out of magic), and you reach the piece of heart on the ledge outside! Be careful jumping off the ledge; the pink flying creature will fly right for you; run to the left immediately after you land to avoid it, then kill it when it pursues.
Now go to where Kakariko Village is in the Light World to find the depressing Village of Outcasts. There are ghosts flying around. There are also rat-faced thieves around who will bump into you, causing you to drop some rupees. Avoid those guys. You'll also see some creepy skeleton-cuccos hopping around; ew.
In the northwest of town there is a house with a treasure chest above the door. Inside the house is an array of 16 treasure chests, and if you pay 30 rupees you are allowed to open two. One random chest contains a piece of heart! Play until you find it. (If you run low on rupees, there are empty houses [the non-destroyed ones] around the village with chests containing 300 rupees each. You shouldn't need to play the game 100 times, though.
Now, we're going to do something a little bit crazy. We're going to do a couple of dungeons out of order! You get such a helpful item in the fourth dungeon that it's worth it to break the rules a little and go there next. The entrance is right here in the Village of Outcasts, so why not?
This dungeon's boss is tough, so be sure to have fairies in your bottles.
Go to where the weather vane was in the Light World to find a creepy statue with a trident pointing down. Looks like there's something behind the trident... pull on it, and it will disappear, revealing steps down. Go down the steps to find...