After you learn Sea Skim in Fuchsia City, you can surf to Cinnabar Island from Pallet Town or Fuchsia City.
Explore Cinnabar Island
In the Cinnabar Island Pokémon Center, there is a Punk Guy named Darko who, if you are playing Let's Go Eevee, will give you an Alolan Meowth in exchange for a Meowth, or if you are playing Let's Go Pikachu, will give you an Alolan Grimer in exchange for a Grimer.
In the Cinnabar Lab, there is a scientist in the Research Lab room who will give you TM02 Taunt.
In the Experimental Lab room, talk to the scientist in the upper-right corner to turn your fossils into pokémon. You got a fossil in Mt. Moon, and you can get another fossil from the side office of the Pewter City Museum.
Cinnabar Island Gym
The Pokémon Gym here is locked, but the note on the door mentions an building overgrown with plants. Go west to find that building. This is the Pokémon Manison. There is an Antidote southwest of the mansion.
When you are ready, go into the Pokémon Mansion.