STORY OF SEASONS: Pioneers of Olive Town

You unlock the fishing pole by entering Manuela/Marcos's house in Olive Town. From here, you can cast your line into most bodies of water and try to catch fish. You can even throw a line into a puddle or small lake. Fishing experience will take a bit to level up, so you'll be catching a surplus for the museum or cooking. If you'd like to track the progress:

1) On the Title Screen, go through the trophy icon to find all tracked trophies. Trophies with cumulative progress can be tracked from here.
2) In-game, go to the Records page of your menu and choose Change Title. For this trophy, you can open the Fishing skill page and see how many you've caught.