It is shown in the second Pokémon movie, Pokemon 2000: The Power of One, that if the three began to fight each other, Lugia, the trio master will rise to stop them, but it would need the help of the Chosen One, who turned out to be Ash Ketchum.
While Moltres was at the Kanto Pokémon Championship, in the GBA games, you can find Moltres in either Victory Road or on One Island. While Articuno is where Ash is with the photographer friend of his, while in the game, it is in Seafoam Islands. While Zapdos is found during Season 5 near a pond, while in the game, you can find Zapdos in the Power Plant.
Their names consist of the first three Spanish numbers, Articuno, Zapdos, and Moltres. Their National Pokédex numbers go in order from uno to tres, and as they are the "three" legendary birds, it makes logical sense. Before each number, there is a word corresponding to the bird's secondary type.