There are four bottles in total, most of them require patience to find. Below, the locations of each bottle, along with detailed instructions of how to get them, is shown.
First Bottle - As young Link, head to Lon Lon Ranch. If you go inside the house, you will see Talon. Play the Super Cucco game, by finding three super cuccos. It may take a while, but don't give up. Be quick about picking up chickens. If you win, you will get a bottle. In addition, you will receive some Lon Lon milk for free.
Second Bottle - As young Link, talk to the Cucco Lady. She lost all of her chickens, and you must find them all. One is located inside a crate. Pick up a chicken, and float to the area where a chicken is on a ledge near a fence by the House of Skulltula. Throw both chickens off the ledge. Pick up another chicken, and go to the entrance of the windmill. Float over to the fence on your left, throw the chicken once you hit the fence, jump down and get the other chicken there. Throw the chickens off the side. Do not leave this area yet. Now, climb the ladder located near the windmill, and find the other chicken there. The rest of the chickens can be found easily.
Third Bottle - This bottle can be found in Lake Hylia. You can either go to Lake Hylia and find it near the stone pillars, or you can find it by Zora's Domain. From Zora's Domain, swim under the water, and find the exit. Go through it, and you'll end up in Lake Hylia. Swim forward a bit (you can see the bottle in front of you), and snatch it. Go back to Zora's Domain through the hole, and give the bottle to King Zora. There is a message inside, he will take the message, and you will have an empty bottle.
Fourth Bottle - This one can be tough to get. As an adult, ride Epona and find 10 Big Poes in Hyrule Field, which you will capture in your bottles. Bring the Poes to the Ghost Shop located in Hyrule Market (near the entrance), and if you capture 10 Big Poes, the owner will give you a bottle. You can only find Big Poes while on Epona, and only in Hyrule Field.