How to get there on the ship?

After you beat the elite four and the credits are finished rolling, you will be able to load up your game again. You will be in your room, at your house in New Bark Town. Head downstairs and your mom will tell you that Prof. Elm was looking for you. Go ahead and stop by his lab and he'll give you the S.S. Ticket!

You can take that over to the dock in Olivine City and then board the S.S. Aqua!

After obtaining the National Dex from professor Oak, you will be able to board the ship after showing the crew member your ticket.

Inside, a man will be looking around in a bit of a panic. Apparently his daughter went missing aboard, and they're not going to be able to set sail until she's found.

While fighting some of the trainers here, If you need ever need healing, you can nap on the bed in your room, which is located in the southeastern most room on the first floor.

After battling any Trainers you want, go down to the basement and try head to the eastern part of the ship. A sailor won't let you pass because his buddy isn't there. He's apparently still sleeping.

Go back to the first floor and inside the room near yours. Wake up the sleeping sailor and he'll run off in a hurry.

Now you can go back to the basement and past the guy over to the engine room. Take the ladder up to the captain's room, where the man's daughter is hiding. Talk to her, and she'll run off.

Chase her down to the basement of the ship, but this time search in the western part of it. You'll see her sort of hiding there. Talk to her and you'll bring her back to her father, who will reward you with the Metal Coat.

Afterwards, the ship will land in Vermilion City.

Hope this helped!