To get to Zora's River, go east (on the map) from the entrance of Hyrule Castle Town and find the river. Follow this south and straight to where it comes from, where you'll the entrance to Zora's River. You'll see Kaepora Gaebora once you get inside, as well as some boulders and a tree.
After listening to Kaepora Gaebora, roll into the tree to make a Gold Skulltula appear. Grab its token, then use a bomb to blow up the boulders in your way. Keep an eye out for the Cucco, then go over and talk to the guy eating beans. He will sell you some Magic Beans. Note that the price for these will rise by 10 Rupees every time you buy one. What you do with them is plant them in the soft soil patches, and they grow into plants that carry Link to hidden places later on in the game.
|There are two new enemies in this area of the world, which are Octoroks and Blue Tektites. Octoroks pop out of the water when you get close enough, shooting rocks at you. These can be deflected back at them with your shield. Blue Tektites are similar to the red ones found on Death Mountain, but they have twice the health and can float on water.|
It's time to grab some Pieces of Heart! Grab the Cucco and start going up the river. When you get to the small incline, jump across the gap and go around the corner to see a Piece of Heart sitting on a small piece of land. What you need to do is with the Cucco in hand, jump to the next ledge and let go of the Cucco just before you reach the ledge. From this ledge, you can do a normal jump to reach the platform with the Piece of Heart.
Once you grab the Heart Piece, jump back down to where the Cucco is and grab it again. Head towards the end of the area, where you'll see a bunch of intertwining paths. Head to the highest point and use the Cucco to float down to the second Piece of Heart. Jump down from the ledge with the heart piece, staying close to the wall, into the water. Play the Sun's Song if it isn't night, which will make a Gold Skulltula appear on the nearby ladder, so kill it and climb the ladder to get the token.
To continue with the story, you should have seen some stone on the ground near the waterfall. While standing on it, play Zelda's Lullaby and the waterfall should disperse around the opening that will lead to Zora's Domain.