Based partly on the fact that raisinghellyer's comment said the answer is "juvenile," I'm going to guess that the answer is
Bulbasaur, Squirtle, and Charmander
They are three, not of a kind, who, in the wild, you cannot find.
The three starters and their evolutions are not found in the wild at all
Against the odds, ever set to rival, you must find your center for survival.
Each of the types is super effective against one other and weak to the other, and your rival always takes the one super effective against yours. The center for survival is a reference to the PokeCenters that can be found in every town/city, where you can heal/revive your Pokemon for free.
Your journey begins, you and one, but the quest is far from done.
You take one started at the very beginning of the game, and continue the game with them.
Countless more may win your heart, but these three, there since the start,
Again, these are the starters. There are at least 148 others in the world, but you will always start with one of these three.
They form you, as you form them, at least until you start again.
Your team is often based around them, though the next time you play through you may pick a different starter.
Don't use a device to learn their names, that's all show and little game.
Pretty sure this is a reference to the Pokedex, though I'm not sure if I would consider that a little game.
Possibly refers to HoHo (credit to Going Hamateur), but again, not that familiar to gens past first