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.
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You can find the Lift Key in the Team Rocket Hideout on Pokemon FireRed. It is dropped by a Grunt on basement floor 4 after defeating him.
use the lift key to use the lift
The lift key is the key to operate the lift in the underground base of Team Rocket in Celadon City. The key can be obtained upon defeating one of the Rockets inside the building. You will need to the key to go to Giovanni, the boss, and when you defeat him, you will obtain the silph scope.
You beat a grunt, and when you beat him he drops the LIFT KEY and you can use it to get to the top floor and face Giovanni ;D