The Lighthouse Keeper needs 6 moths for his first quest he assigns to you, which rewards you with a beginner fishing rod. This guide helps you get moths in the game.
Find flowers to pick at night (they are all over the place) and there is a chance you will find a moth. Try to get at least 6 and then return to the Lighthouse Keeper at day to give him. The quest should be completed and you should collect the fishing rod reward.