To fish, just follow these simple instructions.
Equip a Fishing Rod: You need to equip a rod, either from the menu screen or from your Fast Access Menu. Depending on the rod you use and where you fish, the type of fish you can catch will vary.
Start Fishing: Once you've equipped a rod, just press [[A]] at any fishing spot to start fishing. You'll cast out a float that lets you know if you've got a bite on the end of your line.
Hook a Fish: When you've hooked a fish, the float on your line will sink beneath the water and an "!" icon will flash above your character. Rapidly press [[A]] to reel in the fish. If you don't press [[A]] quickly enough, the fish will get away.
Press [[B]] while standing at a fishing point to jump into the river. While swimming, you may see an "!" flash above your character, indicating that they see something down below. Press [[A]] when you see an "!" to dive down and pick up an item.
If you set up fish hatcheries at harvesting points in the river, you can farm your own fish. After setting one up, wait several days, then return to see what kind of fish are inside. When you harvest fish from a hatchery, you can leave one fish in to allow that same type of fish to multiply. Just be careful; hatcheries do break sometimes, especially if left unattended too long.