When you equip the trident you're able to swim underwater. First return to Purapril by going through the new door in Lilypad. Leave Purapril at the north west exit. You're now outside the castle wall where you've been just after you've left your house. Go to the right and jump in the sea.
Here you can find secret doors to treasure chests, 3 to be exact. The Pygmy Armor is hidden inside the treasure chests, which you have probably seen before. You can reach it by going through the underground maze.
After you've got it, go to Alsedo and make your way to the underground passage. After a short time you'll see a gap filled with water, stand on the bottom and press up to enter a secret door. After getting the heart, go to the end of the passage and jump in the gap filled with water to reach the Monster World Sea.
In this sea you can find another part of the Pygmy equipment. It's located at the right side beyond the cliff. To enter Poseidon's shrine you will need an Amulet. This item is located in the chest at the right end of the Monster World Sea. When you're in the area with the chest and the 4 palm trees, stand between the 4 palm trees and press up, you'll be transported to the Amulet.
Now the entrance to Poseidon's shrine will appear at the bottom right corner of this sea. Once inside swim to the right and hit the red switches with your trident to turn of the vents. After switching them all off go back and go down. When you see a split in the road, take the upper road first to get Return magic. Don't use it yet because it will teleport you to your last saving point.
After that, make your way to Poseidon. He will tell you that the Oasis boots are hidden somewhere inside his shrine. They are in the chest a little bit back which you've probably seen before but were not reach. After getting them use Return to get back to Purapril.