I know there's the PokeBank thing, but from what I've gathered, that will cost a fee to use and it isn't even available yet. Is there any PokeTransfer-like thing in Gen 6 or will I have to wait for PokeBank to become available then pay for that?
You will have to wait for the PokeBank to be released. There is no other way for transfering. Good news is, it will be free for the first month.