Here's the thing: Mr. Mime learns Psychic at level 40 in the main series games, but it cannot learn any moves by leveling up in FireRed.
In FireRed, Mr. Mime can only learn moves through:
* TMs/HMs: It can learn Psychic from TM29.
* Breeding: It can learn Psychic from its parent if it knows the move.
So, if you want your Mr. Mime to learn Psychic in FireRed, you'll need to either get TM29 or breed it with a parent who knows the move.