First, click on the Fishing icon, and pick the island.
(Note: To open new islands, you need to reach the level that the island requires)
Click Chosen Lure
Then select Lure of the type of fish you want and choose Fishing Rod
Move the character closer to the fishing spot; the Fishing button will appear to click on it.
Cast Bar will appear, and you need to try to aim Bobber in Great Zone or Perfect Zone accurately and Tap the button to get a bonus.
Tap - drop continuously to align the Power bar at Sweet Zone (green) to add mana points until the Skill Bar is full, you will be able to use the skill to fish (long press and drag upward)
If you let the Power Bar in Danger Zone (red), you will break your fishing line and fish will fail.
Do not be subjective when you have used skill to fish! You need to pay attention to "fish's HP", if HP is still available, you can still break the fishing line if you don't press the fishing button.
And this's the screen when you successfully fish.