Instructions
Buy the Fishing Rod from Tom Nook's store. Visit his store on another day if he isn't selling the Fishing Rod. You will need the Fishing Rod to find the Golden Key. If you have the Golden Fishing Rod, you can use that instead of the regular Fishing Rod.
Look around town for a neighbor who is pacing anxiously in front of his home. Speak to the animal and he will ask you to look for his lost key. Go to the river nearest to the neighbor's home. You will find every villager's Golden Key in a river.
Equip the Fishing Rod or Golden Fishing Rod. Fish for the Golden Key in the river. At first, you will only catch fish. After you catch about a dozen fish, you will reel in the Golden Key.
Return to the neighbor's home and speak to the animal. Your character will hand the lost key to the animal. She will give you a reward to show her gratitude. The reward might be 600 bells, clothing, furniture or a gyroid.