This page explains how to catch fish in Story of Seasons: Trio of Towns.
Get a Fishing Rod
When you go into Westown on Spring 8 of Year 1, there will be an event where you will get a fishing rod.
Equip your fishing rod when you are near some water. Press A to throw out your fishing line. After a moment, a fish shadow will appear in the water and approach your bait. The fish might nibble on the bait, but don't press A until the bobber goes underwater. Then keep tapping the A button until you catch the fish.
You can buy basic fish bait at various shops in the three towns. Some shops will also have a service to create bait for you from small fish or other ingredients.
Go to Ludus's shop in Lulukoko Village and ask him to attach bait to your fishing rod. The bait will be used the next time you use your fishing rod.
Using a Fish Trap
You can buy a fish trap in Lulukoko and put it into the stone area of Lulukoko beach. When you come back later, there might be fish splashing around. Check on it to take out the fish. The trap has a limited number of uses, but you can buy as many as you want.
Using a Hatchery
You can get a fish hatchery at Ludus's shop and put it on your farm using the work bench. After you catch fish, you can put it into the hatchery to breed more of the same type of fish. You can also get pearl oysters and put them into the hatchery to get pearls.