After you get the Surf Harp from Angler's Tunnel, you get a hint to go to Martha's Bay.
Learn Manbo's Mambo
Outside, swim west from the dungeon entrance and go into the cave that you find there. You encounter Manbo, child of the Sun Fish. If you have an Ocarina, Manbo teaches you Manbo's Mambo. You can use it to warp to Manbo's Pond, which is northeast of the Mysterious Woods.
Help the Ghost
As you explore, 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.
After you take the ghost to the house on the bay, it 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.
After you take the ghost to its grave, it mentions a jar in its house. Then the owl appears and tells you to dive into Martha's Bay to reach Catfish's Maw.
The jar that the ghost is referring to is a jar in its house that you can pick up with the Power Bracelet to find a hidden Secret Seashell under it.
Trade Some Items
You can swim to the bridge east of the mermaid statue in Martha's Bay 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. 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. If you go to the Animal Village and talk to the artist Schule Donavitch, he asks you to put the scale onto the Mermaid Statue that he made. Unfortunately, you can't reach the statue yet.
For now, swim to the middle of Martha's Bay to find the Catfish's Maw. To enter, 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.