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Upon entering the S.S. Anne, you’ll find your rival, Trace, already on board. Not only that, but Blue — Professor Oak’s grandson — is on board as well. He’ll give you a Shalour Sable, which is an item that can heal any status condition.
Now you’re free to explore the ship. It’s quite open-ended, and you don’t have to explore everything, so keep that in mind while following the directions. You can also leave the ship at anytime without worry.
From the entrance, the first room on the right can be used to heal your party to full by talking to the lady at the table. Definitely make use of this whenever your Pokémon need some healing, which they likely will due to how many Trainers are on the ship.
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The second room on the right has Gentleman Arthur inside for you to fight. The third room on the right has nothing for you except for a chuckle. Head south of there to reach the basement.
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The second room from the right has Sailor Leonard inside.
The third room from the right has Engineer Dylan as well as a Full Heal and an Elixir.
The fourth room contains Sailor Huey and a Super Potion just past him.
The final room has just Fisherman Barny in it for you to fight.
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Return back up to the first floor, heal up if necessary, and then head to the west.
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The third room from the right, starting from the entrance to the ship, contains Beauty Nickie and Youngster Tyler, as well as a Paralyze Heal.
On the far western end, you’ll find a staircase going upstairs, but there’s also a path leading south to the kitchen.
Kitchen
There’s not much in the kitchen, although the third trash bin over on the right side of the room has a Tiny Mushroom in it, which can be sold later for 
250. There’s also a Super Repel on the bottom-left part of the room.
Head back to where the stairs leading up were in the previous room to reach the second floor.
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Head to the south and you’ll find some more stairs leading to the ship’s deck. There are several more Trainers for you to fight as well as a few items out there. Otherwise you can skip that and continue going to the right if you’d rather.
Deck
Head along through the corridor until you reach the exit out onto the deck. As you exit, your partner will jump down off of you and look around excitedly.
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1080Fight against the two Sailors on the deck and then head all the way to the west. You’ll have a little scene of you and your partner Pokémon staring out at the water while on the rails, which your partner will enjoy.
There are also 3 Pinap Berries that are hidden by the table on the bottom-right part of the deck.
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From the stairs leading up to the deck, head to the east.
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The second room from the left has Fisherman Dale inside, who has nothing but a Magikarp on his team, making him extremely easy to take down. There’s also a Revive behind him, which you should pick up.
The fourth room from the left has Gentleman Brooks and Gentleman Lamar in it, who both give you a pretty good amount of money for beating them, but on top of that there’s a Nugget on the chair in the room which can be sold for even more.
None of the other rooms have anything worth picking up or other Trainers to fight, so there’s no need to check them out.
Make sure you’re healed up before proceeding, because you’ve got another fight against your rival, Trace, who somehow got ahead of you as he came down from the stairs leading up to the third floor’s captains quarters.
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His Pokémon have gotten quite a bit stronger since the last time you’ve fought him back in Cerulean City, although they shouldn’t be too bad. He has upgraded to a Super Potion in battle, which he’ll use when his Pokémon get weakened enough.
After beating him, he’ll leave, telling you to meet the captain to learn the Secret Technique. Continue up the stairs to do just that, after optionally shaking your Joy-Con to see a little scene of your partner happy that it has beaten your rival.
Captain’s Quarters
Talk to the captain, who is suffering from severe seasickness. You’ll give him a back rub, which somehow helps with that.
As thanks, he’ll volunteer to teach you his Secret Technique, but then your partner Pokémon will show interest in learning it instead.
After a little bit of a demonstration, your partner will learn the Secret Technique, Chop Down, which allows you to finally cut down those small trees you’ve seen blocking your path at various points in the game.
At this point, make sure you’ve gotten everything you want from the S.S. Anne and fought whichever Trainers you wanted to, because you won’t be able to return after leaving.
Leaving the S.S. Anne
As you leave the S.S. Anne, you’ll witness a cutscene of the S.S. Anne departing, sailing away, never to be seen again.
Before you leave the area, though, there’s a girl you can talk to near the railings. Talk to her twice and she’ll offer to battle you on the second time talking with her.
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She uses Fairy-type Pokémon on her team. The first she uses, a Jigglypuff, shouldn’t be too rough, but her Mr. Mime can actually be pretty resilient to Special-based moves and may prove to be more of a challenge.
She’ll reward you with a Bottle Cap after beating her in battle, which won’t have any use until after beating the game, but they’re still very good to have. You can also challenge her once per day, earning another Bottle Cap with each victory.
As you head back towards Vermilion City, you’ll see the Gym Guide staring at the small tree you can use Chop Down on. Heal up your Pokémon if necessary and then make your way over there. You can interact with the tree to get a prompt asking if you’d like to use Chop Down to chop it down.
After using it, you’ll chop it down, allowing you access to the Vermilion Gym! There are, of course, plenty more of these small trees that can be chopped down.
Most of the optional ones are back on earlier routes, which you won’t be able to access until after beating the Vermilion Gym, so don’t worry too much about backtracking for right now to get them. Instead, for the time being, get ready to take on the third Gym in the game!