The metagame in general is going to slow down significantly as Fairy types get immunity to dragon type moves and are super effective against Dark and Fighting. Outrage sweepers are going to have a much more difficult time plowing through teams, and Kingdra will probably see a sharp drop in usage (maybe even a drop to RU). I imagine that Fairy dual typings will make the biggest difference, as something like a Grass/Fairy type could be a fantastic counter to Pokemon like Garchomp.
Fairy types are also weak to Poison and Steel so I imagine players would change their movesets accordingly to use poison/steel more often as general coverage moves. One example I can think of is with Toxicroak- if there are a lot of tanky Fairy types, I might change Ice Punch with Poison Jab on the standard Swords Dance set to help smash through them. Since dragon types would be less ever present (and more easily dealt with by Fairy types of my own), the Ice typing might not be worth it. This of course is entirely dependent on the stat distributions of the new Fairy Pokemon.
At the very least, we will likely see a lot of Fairy type coverage moves used as a replacement for Ice type moves, especially the currently omnipresent Hidden Power Ice. If Sylveon could run a set like Moonblast, Thunderbolt, and Hidden Power Fire, I imagine it could be an extremely effective revenge killer for sun teams. We'll see if weather even has a major role in Gen 6 though, the whole permanent weather mechanics may be revamped entirely.
I would like to point out that Fairy type is super effective against Dark, then Hydreigon has a 4X weakness to Fairy attacks. Fairy is also super effective against Fighting, then Scrafty also gains a 4X weakness. That's 2 Pokemon commonly seen in VGC gaining a 4X weakness. That should change the metagame fairly drastically.
Or should I say: "That should change the metagame fairyly drastically?"