You can get a fishing rod at Marcos's house, then use the fishing rod when facing a body of water.
How to Get a Fishing Rod
To get a fishing rod, go into Marcos's house in the southeast of Olive Town. You will get a fishing rod the first time you visit.
How to Fish
To use the fishing rod, move it from your tool bag to one of your item bars, then select it, and face a body of water. Press Y to cast the line. Wait for an exclamation mark to appear above the bobber, then press Y to hook the fish.
If the fish in the gauge is white or blue, hold down the Y button to reel it in. If the fish in the gauge turns red, it means that the fish is angry and will reduce your durability quickly, so let go of the Y button when the fish turns red. When the fish turns white or blue again, hold down the Y button. Repeat the process until you catch the fish.