Get into the water as a human and dive. As Link touches the bottom, put on the zora mask. You will notice that the action on B has changed from "punch" to "dive". Dive and you can now swim as if in normal depth water to the Big Octo.
You will notice after using swimming in shallow water, you cannot pull out your zora fins to kill the Big Octo. Instead of taking off the zora mask and putting it back on, you can get close to a wall or tree, and press A to swim. You will run into the wall and store a ground jump. From here, you can simply attempt to backflip, then pull out your fins and kill the Big Octo.
Instead of having to go out the cave by the deku palace and navigate past the dragonflies, turn into a deku and get onto the Lily pad nearest to the rocky outcrop. Aim yourself towards the downward slope and turn into a human. Quickly roll, and hold forward on the control stick to ensure that you land on the outcrop. Now you can simply use the deku flower to fly directly to the song stone.
Normally when you go to get the pictobox from the witch at the swamp tour shop, you have to take a trip on the boat to the deku palace. However, you can skip riding the boat by mashing the ocarina button as soon as you recieve the pictobox. If you have trouble with this, you can instead mash the pause button, then unpause and pull out the ocarina. From here, play the Song of Soaring.