Anju (or, “Chicken Lady”, as she was affectionately referred to before the release of Majora’s Mask) has lost her seven Cuccos. If you help her catch them, she’ll give you your first bottle as a reward.
Talon has a game in the day where you have to find three special Cuccos that he throws in amongst all his other Cuccos. It’s hard, but if you clear the area by throwing the Cuccos behind Talon before starting the game, it’s a little easier.
The third bottle is at the bottom of Lake Hylia. This bottle is necessary if you want to get into Jabu Jabu’s Belly, and you can find it by the Lake Hylian entrance to the Lake-Zora’s Domain shortcut, which is near the shoreline. Just dive down to get it.
As an adult, visit the Guard House in Hyrule Castle Town, which is now occupied by a Poe Trader. To get this bottle, you need to collect ten Big Poes from Hyrule. They only appear when you’re on Epona, and once defeated, you need to bottle their souls and sell them to the Poe Trader. Here are there locations:
If you want a fifth, or even sixth bottle, and you have an unneeded item like the receipt for Biggoron’s Sword, then go out with a normal bottle and swipe to catch something. Before you actually get it, press start and swap the bottle with something else (like the receipt). The item will be replaced by a bottle, and you can keep anything in it. However, make sure you will never need the replaced item again because once you have swapped it, it’s gone forever.