In Pokemon FiredRed, players can only get Mew directly from Nintendo via the company's promotional events or by trading with another player who has one. Therefore, Mew cannot be caught or obtained in the game by conventional methods.
Mew is a psychic-type legendary Pokemon. It is one of the rarest types of Pokemon and the last one in the Kanto Pokedex at #151, after Mewtwo. Mew's DNA was used to create Mewtwo. The entry of Mew on the FireRed Pokedex version states, "A Pokemon of South America that was thought to have been extinct. It is very intelligent and learns any move."
Mew indeed has the ability to learn every TM, HM and Move Tutor because it is said to possess the genes of all Pokemon. However, it cannot learn abilities that are exclusive to a particular Pokemon or Pokemon group. Mew also has the ability Synchronize, a rare move that passes the status conditions inflicted on it back to its opponent. Mew has no gender and no evolutionary chain.
Outside of Nintendo sponsored events, a player may use an Action Replay cheat code in order to get a Mew on Pokemon FireRed. The downside to obtaining a Mew this way is that the Pokemon will often disobey and it can't be fully controlled. If the player's only objective in trying to obtain a Mew on FireRed is to fill up slot #151 on the Pokedex, then this is a suitable method.