In Vermilion City, you can go south to the port and board the cruise ship, S.S. Anne.
Aboard the S.S. Anne
Go south in the ship and you will find your rival, and Blue, who gives you a Shalour Sable. Blue warns you about Team Rocket, then leaves.
In the room near your rival, you can talk to the woman to heal up your pokémon.
In the room to the right of that, you can battle Gentleman Arthur, who has a level 18 Growlithe. You get 2160 Pokédollars and 3 Poké Balls for winning.
Go south from here and go down the stairs.
In the second room, you can battle Sailor Leonard, who has a level 18 Slowpoke. You get 1080 Pokédollars and 3 Poké Balls for winning.
In the third room, you can battle Engineer Dylan, who has two level 18 Magnemites. You get 1080 Pokédollars and 3 Poké Balls for winning.
In Dylan's room, there is a Full Heal and an Elixir.
In the fourth room, you can battle Sailor Huey, who has a level 18 Psyduck and a level 18 Poliwag. You get 1080 Pokédollars and 3 Poké Balls for winning.
In Huey's room, there is a Super Potion.
In the fifth room, you can battle Fisherman Barny, who has a level 18 Goldeen and a level 18 Horsea. You get 720 Pokédollars and 3 Poké Balls for winning.
Go back the way you came and go up the stairs. Go west from your rival.
In the third room to the left of your rival, you can battle Beauty Nickie, who has a level 17 Pidgey and a level 17 Pikachu. You get 1020 Pokédollars and 3 Poké Balls for winning.
Also in this room, you can battle Youngster Tyler, who has a level 16 Mankey. You get 320 Pokédollars and 3 Poké Balls for winning.
Also in this room, there is a Paralyze Heal.
Exit that room, then go west. Don't go up the stairs yet. Instead, go south to reach a kitchen. In the southwest corner of the kitchen, there is a Super Repel. Check the southeast trash can (between two of the chefs) to find a hidden Tiny Mushroom.
Now leave the kitchen, go north, and go up the stairs.
Go south, then go up the stairs to the left.
Keep going west to go outside.
Go west and you will battle Sailor Trevor, who has a level 18 Seel and a level 18 Machop. You get 1080 Pokédollars and 3 Poké Balls for winning.
Southwest of Trevor, you can battle Sailor Edmond, who has a level 18 Poliwag. You get 1080 Pokédollars and 3 Poké Balls for winning.
Check on the table to the southeast to find 3 hidden Pinap Berries.
Go to the left to see a cutscene.
Go back downstairs, then go east to go down the next set of stairs. Go east, and in the second room, you can battle Fisherman Dale, who has a level 18 Magikarp. You get 720 Pokédollars and 3 Poké Balls for winning
North of the Fisherman, there is a Revive.
Exit that room and go east. In the fourth room, there are two Gentlemen. Here you can battle Gentleman Brooks, who has a level 18 Ponyta. You get 2160 Pokédollars and 3 Poké Balls for winning.
North of Brooks, you can battle Gentleman Lamar, who has a level 18 Sandshrew. You get 2160 Pokédollars and 3 Poké Balls for winning.
North of the Gentlemen, there is a Nugget.
Leave that room, then go east, then north toward the stairs. You will battle your Rival, who has a level 20 Pidgeotto, a level 20 Oddish, and a level 21 Pikachu or Eevee (a Pikachu if you are playing Let's Go Eevee, or an Eevee if you are playing Let's Go Pikachu). You get 1260 Pokédollars for winning.
After the battle, go up the stairs and talk to the Captain. During the cutscene, your partner pokémon will learn the Secret Technique Chop Down.
To leave the ship, go downstairs from the Captain's quarters, then go south, then west, then north, and go down the stairs. Then go east, then north to leave the ship.
Outside, there will be a cutscene.
Back in Vermilion City
Afterward, go north and talk to the person looking out at sea. This is Mina, who was in Pokémon Sun, Pokémon Moon, Pokémon Ultra Sun, and Pokémon Ultra Moon. After you talk to her, you can talk to her again to battle her Fairy-type pokémon. She has a level 21 Jigglypuff and a level 22 Mr. Mime. You get 2640 Pokédollars and a Bottle Cap for winning.
Go north. The guy near the Vermilion City Gym will signal to you, to remind you that you can cut down the tree next to the gym now.
When you're ready, go to that tree next to the Vermilion City Gym and check on it to chop it down, then go into the gym.