To get the Fly move in "Pokémon FireRed," talk to the recluse living in the house at the far north of Route 16. You need to use the Cut move on the tree blocking the north gate in order to get to her. She gives you HM02 and asks that you keep her location a secret.
In order to teach your Pokémon to Fly with HM02, you need to have the Thunder Badge from beating Lt. Surge in Vermilion City. Once a Pokémon learns Fly, it cannot forget it to replace it with another move, and it can only be removed by a Move Deleter.
Using the Fly move outside of battle lets you travel to the Pokémon Center of any town you have already visited, but you must be outside to use it. A Pokémon can use Fly in this manner even if it is knocked out. In battle, Fly is a physical attack during which your Pokémon flies into the air one turn and then flies back down to do damage the next turn. During the turn when it is flying, the Pokémon cannot be hit by any attacks except by the moves Gust, Thunder, Twister, Whirlwind or Sky Uppercut.