I'm seeing a lot of speculation in here that goes against what we already know, so let's separate this in to facts, and speculation. The facts:
>10% form is usable on its own. You can either have "regular" 10% with Aura Break or "complete" 10% that has the special ability
>Conversely Complete form is only usable, barring some special plot related event, as an in-battle Pokémon from either 10% or 50% transforming in to it
>The cores and cells are confirmed to not be usable in battle whatsoever. For all intents and purposes consider them like items.
>Very slight speculation, but it's fairly confirmed that there are between four and ten cores
All of this can be confirmed through what has been officially revealed, I can cite sources/go in to more detail if you want on any of those.
SPECULATION:
>You will be given/led to one core, possibly it is what's in Lillie's bag.
>If Zygarde is going to be part of the main plot, you will be led to/given the other cores.
>If he's not (let's hope!) finding all the cores and cells scattered throughout the islands will be a side quest/the post game. You may or may not be given a map with their locations
>Going in to wild, fanfic-tier rampant speculation here, but if you collect enough cells/cores you will be able to use Zygarde's complete form at any time, not just in battle.
