After traveling through the Underground Path and exploring Route 6, you reach Vermilion City.
Explore the City
To the left of the entrance to Vermilion City, there is a police officer who will give you a Squirtle if you have caught at least 60 pokémon.
In the Pokémon Center, a Hiker will give you an Alolan Geodude if you give him a Geodude.
In the northeast corner of the city, there is a hidden Full Heal to the left of one of the orange crates.
In the Pokémon Fan Club building (south of the Pokémon Center), listen to the chairman and he will give you a clothing set in the style of your partner pokémon (an Eevee set if you are playing Let's Go Eevee, or a Pikachu Set if you are playing Let's Go Pikachu).
The woman on the bench to the right of the Pokémon Fan Club will ask you a question. Answer it, then if you catch five of the pokémon she asks for, come back and talk to her to get a pokémon. She will ask for five Meowth if you are playing Let's Go Eevee, and will give you an Arcanine. She will ask for five Growlithe if you are playing Let's Go Pikachu, and will give you a Persian. If you put Arcanine or Persian in your party and take it out of its pokéball (an option when you check on a pokémon in the Party menu), you will ride around on that pokémon! It is much faster than walking! However, your partner pokémon won't wiggle its tail near hidden items while you are riding around on a pokémon.
There are other rideable pokémon, too. Check the List of Rideable Pokémon for more information. Note that some pokémon are faster than others.
Detour to Route 11
Before you board the S.S. Anne, you can optionally go east to Route 11 for some trainer battles, pokémon, and items. You can also get a new feature for your Pokémon Box if you go there.
Go to the south part of Vermilion City to find a wooden pier. Keep going south on the pier and a sailor will give you a Sailor Set for you and your partner pokémon.
Keep going south to board the S.S. Anne.