Zora's Domain:

The first thing to do here is get an empty bottle and find the little fish floating around in the shallows near a circle of stones.  Use the bottle to catch one of the fish.  You'll need it later.  Now head up to the top of the mountain in the middle of the domain and you'll see King Zora.  Before doing anything else, take this opportunity to use one of his torches to light a Deku stick and quickly run and light all the other torches in the domain, including two hidden behind a waterfall.  This opens a few things up for you, like a heart container piece. Go back to the King's room and talk to him, then head left from his room and find the diving game. Win at the diving game and you'll get the Silver Scale, which allows you to dive deeper.  Now look for an underwater doorway at the front of the mountain.  Dive down and go through it and you'll pop up at lake Hylia.  Swim forward a bit and you should notice a bottle at the bottom of the lake. Dive down and fetch it.  Then swim back through the tunnel to Zora's Domain.  Bring the bottle to King Zora and show it to him (press C when Z Targeting him) and he'll take the note out and move out of your way.  Now take the right path to get behind him and enter Zora's Fountain.

Zora's Fountain:

Ignore the big fish (Jabu-Jabu) for now and instead swim behind him to the southern part of Zora's Fountain to a small island with a tree, some bushes, and two boulders.  Put a bomb next to the gray boulder to blow a hole in the wall and get the Farore's Wind spell from a great fairy.  Now swim onto the platform by Jabu-Jabu's mouth and walk up to him.  Use the bottle with the fish in it to offer the fish to Jabu-Jabu, thereby making him open his mouth and swallow you.

Jabu-Jabu's Belly:

Use your slingshot to shoot the white thing hanging down from the ceiling and unblock the door.  Go through the door.  In this room you'll first meet the electrifying jellyfish.  Don't hit them with you're sword, use a Deku stick instead.  You'll break the stick but you'll at least have killed them (this is the only time I know of where one of these sticks is at all useful for anything other than lighting torches and spider webs).  Now follow the path around and IGNORE the elevator platform and the alcove in the wall (you'd be surprised how many people go for this and get all mixed up).  Instead go through the next door and in this room, the pit room, you'll meet Ruto.  She'll jump down a pit...follow her down.  Talk to her twice and she'll demand you carry her.  You'll have to do so.  I know it's a pain and it may seem easier to just leave her there and go get her at the end, but you'll need her later in the level.