Walkthrough
After the Main Story Credits are finished, the Post Game will officially begin with your trainer back in Pallet Town. By leaving your home, you will run into Professor Oak. If you have captured over sixty different Pokemon Species, he will upgrade your Kanto Pokedex to the National Pokedex. This is essential to have for the Post Game Story, as you won't be able to complete it without the National Pokedex. If you haven't captured since many Pokemon Species, do so before doing anything else so Professor Oak can give you the National Pokedex.
With the National Pokedex in hand, you will actually have the choice of what to do next in your journey. You can either go to Cerulean Cave on the outskirts of Cerulean City - now accessible since your trainer is the Kanto Region Champion - in order to catch the strongest Legendary Pokemon found within the game. Or, you can head back to One Island by using the ferry in the Vermilion City Harbor in order to help Celio finish building his machine.