After you've reached and explored Kiloude City, you must complete at least one round in the Battle Maison in order to continue. The battles in here start off very easy, so it should be no trouble. In the back room, talk to the receptionist at the right to get started. Choose "Challenge" and select whichever battle format you're comfortable with. Your Pokémon, as well as your opponent's Pokémon, will all be set to level 50 for these battles only (your Pokémon will go back to normal as soon as you leave).
After winning the first battle, choose "Interrupt" if you don't wish to keep going (you only need to complete one battle in order to progress with the story). When you're ready, exit the Battle Maison, and head to the north of Kiloude City, to the area with the Artist. Your rival should now be standing here, ready to battle you.
Meowstick L66 is out first, unsurprisingly, ready to use Fake Out on the first turn. It also has Psychic, Shadow Ball, and Disarming Voice as damaging moves.
Her new addition, a Clefable L68, has the moves Moonblast, Focus Blast, Meteor Mash, and Sing. Remember that Clefable is pure Fairy-type, so Poison- or Steel-type moves work well here. Also keep in mind that it has the Magic Guard ability, which means it can only take damage directly from moves. Indirect damage, such as from entry hazards, status ailments, or weather conditions will not hurt it.
Her Altaria L67 isn't much different than last time, still packing Dragon Pulse and Dazzling Gleam to damage you with. Remember that Ice-type moves are extremely effective against it!
Her eeveelution isn't much different either, though its move of the same type has been replaced by a tougher (yet more inaccurate) move of the same type. If you started with Chespin, she will have her Jolteon L66 who now knows Thunder. If you started with Fennekin, she will use her Flareon L66 who has learned Fire Blast. If you started with Froakie, she'll have her Vaporeon L66 with the newly learned Hydro Pump.
Her starter, like the eeveelution, has also learned a tougher move since last time. If you started with Chespin, she will have her Delphox L70, who now knows Fire Blast, and still knows Psychic and Shadow Ball to damage with. If you started with Fennekin, she'll use her Greninja L70, who can use Hydro Pump in addition to Dark Pulse and Ice Beam. And finally, if you started with Froakie, she'll send out her Chesnaught L70, who knows Wood Hammer, as well as Brick Break and Body Slam. Remember that no matter what starter she uses, its ability will make its new Fire, Water, or Grass move even more powerful if its HP drops below 1/3.
Finally, out will come her Absol L68. Her Absol is far from the same as before. It has the same moves (Night Slash, Quick Attack, Slash, and Swords Dance) but it can now Mega Evolve! Once it Mega Evolves, Absol's already high Attack raises even further, and its Speed gets drastically better, too. It's still very frail, though, so if you can manage to hit it with something decently strong, you shouldn't have much trouble.
After the battle, your rival will give you an Absolite, which is the very Mega Stone used to Mega Evolve Absol. She will then explain that Professor Sycamore may have discovered something about Mega Evolution, and suggest you head to Anistar City to check it out. Go to the sundial in Anistar City, where the Professor is waiting.
Speak to Sycamore at the sundial, who will begin talking about his theory on Mega Stones, and how it connects to Anistar's sundial. He will explain that the sundial starts to emit light for one hour every night starting at 8pm. Then, he will instruct you to touch the sundial, which causes your Mega Ring to upgrade.
After you finish this scene, you will be able to search for new Mega Stones all around the Kalos region, but only between 8pm and 9pm. I will be covering where each and every Mega Stone is located in the next part.