After you get the Surf Harp from Angler's Tunnel, you get a hint to go to Martha's Bay.
Get a Piece of Heart
Swim east from the Angler's Tunnel dungeon entrance and go into the cave. Swim to the center of the room and press B to dive. You will find a Piece of Heart.
Learn Manbo's Mambo
If you got the Ocarina earlier, swim west from the Angler's Tunnel entrance and go into the cave that you find there. You encounter Manbo, child of the Sun Fish. If you have the Ocarina, Manbo teaches you Manbo's Mambo. (If you didn't get the Ocarina yet, visit the Dream Shrine in Mabe Village to get it.
In the Game Boy version, you can play Manbo's Mambo on the Ocarina to warp to Manbo's Pond, which is northeast of the Mysterious Woods. In the Nintendo Switch version, you can play Manbo's Mambo to warp to any warp point you have activated. In both games, you can use it in dungeons to warp to the dungeon entrance.
(Switch Version) Secret Seashell and Piece of Heart
In the Nintendo Switch version, you can get a Secret Seashell and a Piece of Heart nearby. From the warp point east of the keyhole that unlocked Angler's Tunnel, go south, then west, then south down the stairs into the water. Swim west until you see a waterfall. Dive underwater at the base of the waterfall to get a Secret Seashell. Then swim south and follow the path until you are near the bridge to Kanalet Castle. Look for the Piece of Heart underwater and dive there to get it.
(Both Versions) Secret Seashell
If you go to the lake north of the hedge maze near Richard's Villa, you can now swim to the island in the northern part of the lake where there is a bush. Cut it to get a Secret Seashell, but in the Game Boy version, be sure to cut the bush from above so the Secret Seashell moves downward instead of upward into the lake. (If it falls in the lake, just leave the screen and come back and cut the bush from above this time.)
Help the Ghost
After you walk/swim around for a while, a ghost will start following you. It wants you to take it to the house by the bay. This house is east of the beach where you found your sword. If you find an unopened chest along the way, open it to get rupees.
Go into the house east of the beach. Afterward, the ghost asks you to take it to its grave. The grave is northeast of the village, and west of the witch's hut where you got the magic powder. Manbo's Pond is close to it, so you can play Manbo's Mambo to warp there, then go south until you reach the witch's hut, then go west, lift the boulder, and go down the ladder.
In this area, cut the bush and use the shovel to dig where the bush was. You will find a Secret Seashell.
Go west and north to reach the ghost's grave. After you take the ghost to its grave, it mentions a jar. In the Nintendo Switch version, you will get a Fairy Jar. Then the owl appears and tells you to dive into Martha's Bay to reach Catfish's Maw.
In the Game Boy version, walk up to the grave after you take the ghost there to get your photo taken there.
There is a jar in the ghost's house that you can pick up to find a hidden Secret Seashell under it.
Trade Some Items
If you don't have the Fish Hook, you can go to the Animal Village and give the Hibiscus to the goat who is writing a letter. Then take the letter to Mr. Write, who is north of the Mysterious Forest. He gives you a broom. Give the broom to the old lady in front of Ulrira's house. She gives you a Fish Hook. In the Nintendo Switch version, you can go back to Animal Village and talk to the goat who gave you the letter to get a Secret Seashell.
Buy the Bow
If you haven't bought the bow from the shop in Mabe Village yet, and if you have enough rupees, it is recommended to buy it before the next dungeon.
Find Items Around Martha's Bay
You can find Martha's Bay between Animal Village and the ghost's house. There are several items that you can find in the area. In the southeastern part of the bay, there is an owl statue that you can reach by crossing a bridge. Dig with the shovel to the left of the statue to find a Secret Seashell.
To the southwest of the bay, there is a phone booth, and south of that, you can swim in the water to reach an island with a bush on it. Cut the bush to get another Secret Seashell.
East of there, you will find some holes and some stairs beyond the holes. If you bought the bow from the Mabe Village shop, you can equip both the bow and the bombs and press both buttons at the same time to shoot a bomb at the bushes. After that, you can use the Pegasus Boots and Roc's Feather to jump over the holes, cut the bushes, and go down the stairs. Go through the cave, and in the Nintendo Switch version, dive underwater to get the Piece of Heart there. At the end of the path, you find a well where you can sprinkle magic powder to make the Mad Batter/Li'l Devil appear. He will increase your capacity for Bombs, although if you say no, you can choose to increase the capacity of Magic Powder or Arrows instead.
In the Nintendo Switch version, south of the Mermaid Statue in Martha's Bay, you can dive underwater to get a Piece of Heart.
Get the Scale
In Martha's Bay, you can swim to the bridge east of the mermaid statue and dive at the spot of the bridge that isn't blocked by rocks. You will find a fisherman on a boat under the bridge. Use the Roc's Feather to jump onto his boat. Give him the fishing hook that you got from the trading sequence. You will get a necklace. Find the mermaid swimming in Martha's Bay and give her the necklace. She will give you a Mermaid's Scale. You can't reach the Mermaid Statue (which is where the artist Schule Donavitch in Animal Village says to put the Mermaid's Scale), so you'll just have to hold onto the scale for now.
Then swim back to the center of the bay. To enter Catfish's Maw, go between the rocks on the bottom-left of the Catfish's Maw, then dive. Swim underwater to the east, then go up to the top of the area. You will emerge in the middle of the rocks, and you can swim into the Catfish's Maw.