To cast your fishing line, look in your pockets and drag the fishing rod to the picture of your person. This will equip the fishing rod.
When you press the A button while holding the rod, you will cast the fishing line in front of you, a fixed distance away. If you are not standing close enough to the water for the bobber to land in it, your person will frown and pull the fishing line back. But if you aimed correctly, the bobber will land in the water. You can also tap a spot with your stylus where you want the bobber to fall.