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I'm admitadly late to the party here, but with a new type coming into the picture for the first time in over a decade, I thought we could discuss what it might possibly bring to the formula.
What we know: Fairy will be strong against dragon. That's about it for what was confirmed.
Whether or not it will resist dragon, or even be imune, who can say.
Some other types I'd like to bring into consideration.
Fighting type.
I think they should be weak to fairy. The last time new types got added to the the mix, Fighting ended up being strong against both of them, bringing them to a total of 5 types they're strong against. 5! That's over a quarter of the types, even counting Fairy. I think they could stand being taken down a peg.
Psychic
I feel the oposite as to fighting. Since unlike the former, the last 2 new types nerfed them rather heavily, particularly Dark type, which was outright immune to them. If fairy's going to have a weakness, psychic seems like a good choice to me.
Ghost and Dark
I feel one of these should be weak to Fairy, and one strong against it. Leaning towards Dark being strong(Fairies wouldnt like the dark, right?), and Ghost being weak to it, mostly becaause of it's 2 immunities.
Against itself
will Fairy join Ghost and Dragon as types weak to itself? Maybe, depends on how many other types they decide to make it weak against. I don't really see it happening though.