Where do you get the Legendary Pokémon in Pokémon Red?
Arcticuno: on the Seafoam Islands
Moltres: in the Victory Road
Zapdos: in the Power Plant
Mewtwo: after you beat Elite Four go to the Cerulean Cave, before the Elite Four was beaten, it will be blocked by some guy.
Mew: Look in the cheats.ign.com for the Mew cheat, it should be there.
If you mean the Red Gyrados as in the one from Gold Silver and Crystal that you were 100% guaranteed to come across through normal playing then there isn't one. The only possibility of finding a red Gyarados is just being lucky enough to come across either a shiny Magikarp, catching it and then evolving it or catching a shiny Gyarados.