You have to battle all those Team Rocket grunts first. The Last One will drop the lift key.
You don't
The key is hidden in the basement of Pokemon Mansion which is located on Cinnibar Island.
To get the lift key, talk to one of the Rocket trainers you have already beaten. He will drop the key, then you can pick it up.
The key to the gym
use the lift key to use the lift
the key is in the basement
you get it by searching for it for yourslf
he's on the bottom floor but you need to take the elevator on the 1st or 2nd floor ( you have to have the lift key) and take it to the bottom floor
i think it's at the basement floor on a table.
it's in the instruction booklet
The key is in the Pokemon mansion.
The Secret Key that you get in the Pokemon Mansion is used to enter the Cinnabar Gym.