Once you set sail to the bottom-most island on the righthand side of the world map (dead tree and cauldron looking icons above it), you will find a fishing rod near the left side of the island that must be unlocked using gems. After you have unlocked the rod, you can pay a single coin to cast the line. The line will remain still for a few seconds and will then bob down letting you know that you've hooked a fish. You must pay another coin to reel the line back in, otherwise you will lose the fish.
There are 8 fish in total, and they are all displayed just to the left of the fishing rod showing you which ones you've caught thus far. The one that took the longest was the bottom left one on the display. Thankfully, you shouldn't be wasting hours of time on annoying digital fishing, as I was only half paying attention and managed to get all 8 unique fish types in 10-15 minutes (maybe less, wasn't keeping track of time).