Quick answer: Get the Basic Rod from the twins' quest (Twins), put it on your tool belt (not the backpack), and go to a marked fishing spot — standing by any water doesn't work. You need Farming mastery 1. Make bait at the Kitchen Table: Breadcrumb, Insect or Live Bait. When the yellow ! appears, hold to pull; when it disappears, tap instead of holding, or the line turns red and snaps.
This page covers fishing, how to get bait and fish respawn.
Where to fish
Fishing spots are in the Village, the Forest river, the Swamp and the Port — see the Map.
Steps
Get the rod
Finish Two Peas in a Pod (6 Clay Plates).
Equip it on the tool belt
Carrying it in the backpack isn't enough.
Make bait
At the Kitchen Table. Matching bait to the spot improves the catch.
Reel it in
Hold on the yellow exclamation mark, tap when it vanishes, never hold while the fish fights.
Sources: Prima Games, wikily.gg, Fextralife.