When you have defeated the Elite 4 and Gary, Prof. Oak will come and congratulate you, your Pokemon will be entered in the Hall Of Fame. You will see a message saying: Entering Pokemon into Hall of Fame SAVING DO NOT TURN POWER OFF.
Don't turn the power off, you will be VERY sorry if you do! You must wait for the saving sequence to finish.
After that you will see the credits. The credits are somewhat long so you can go ahead and turn off the game. Whether you turn off the game of decide to watch the credits, next time the game starts you will be in front of your house in Pallet Town.
Prof. Oak will come to see you, and if you have caught 60 Pokemon he will give you the National PokeDex. If you do not have at least 60 Pokemon (this happened to me once) he will tell you to come back once you have captured 60 species.
Now its the time to go exploring the Sevii Islands. Go to Vermilion City to take the ferryboat. BUT WAIT! If you left Bill waiting for you in Cinnabar Island (yes you can leave him waiting and beat the Elite 4 if you want) then you can not go to Vermilion fort yet. Go to Cinnabar and find Bill in the Pokemon Center, he will give you Tri-pass for the Islands.
Good Luck!!!! :)