After you become the Champion, Professor Sycamore gives you the TMV Pass so you can reach Kiloude City.
How to Get to Kiloude City
After you defeat the Elite Four and the Champion, go to Lumiose Station on North Boulevard of Lumiose City. Talk to Professor Sycamore there and he will give you the TMV Pass. Check on the orange ticket machine next to the train to ride to Kiloude City.
After the train drops you off, talk to the girl on the left to take a pokémon quiz. If you get the correct answer, she gives you a Max Revive.
Go outside. An old guy gives you the Vs. Recorder. You can use it to record videos of your battles. You can upload them to the Internet and search for other players' videos as well.
In the Pokémon Center, a purple-haired guy on the left will judge the potential of any pokémon in your party.
In the house east of the Pokémon Center, the girl upstairs will give you TM91 Flash Cannon.
Go up the stairs east of that house, then go east to find a Nugget.
Go west from there to find the Battle Maison. In the Battle Maison, you can battle many trainers, but you earn BP instead of Exp. You can exchange the BP for items, such as EV-boosting items, trade evolution items, and TMs.
Be sure to win at least one battle in the Battle Maison now. You have to do this in order to get your Mega Ring upgraded. You can interrupt the challenge after one battle. The next time you visit the Battle Maison, you can pick up where you left off.
Battle Your Neighbor
After you win at least one battle in the Battle Maison, go to the north end of Kiloude City. You will find your neighbor. Talk to your neighbor to do battle. Your neighbor has a level 66 Meowstic, a level 68 Clefable, a level 66 evolution of Eevee (Vaporeon if you chose Froakie, Leafeon if you chose Chespin, Jolteon if you chose Fennekin), a level 70 starter that is strong against your starter, a level 67 Altaria, and a level 68 Mega Absol.
After the battle, your neighbor gives you Absolite and says to meet Professor Sycamore at the sundial of Anistar City.
Go to the Anistar City sundial. The professor is standing in front of the giant pink crystal. He'll explain his theory about Mega Stones and the Mega Ring, and he'll ask you to touch the crystal. Just press a button to touch it. After that, you can find hidden Mega Stones from 8PM to 9PM. Check the mega evolution page for a full list of Mega Stone locations.
Continue Exploring Kiloude City
Back in Kiloude City, in the house west of the Battle Maison, Ace Trainer Anton asks you a bunch of questions about Levitating pokémon. All of the answers are correct. Then he asks if you want to battle his Levitating pokémon. He has a level 65 Weezing, a level 65 Bronzong, and a level 66 Eelektross.
North of Ace Trainer Anton's house, in the left-hand house, there is a girl at the table on the right who will give you TM58 Sky Drop.
If you go all the way east from that house, you will reach the Friend Safari. Use the Dowsing Machine outside of the Friend Safari to find a hidden PP Up on a bush.
In the Friend Safari, you can go on a safari in which you can catch different types of pokémon. There is a Safari for each person you have registered in your Nintendo 3DS Friends List. Each friend's safari features a specific type of pokémon, and each safari can have up to three pokémon in it. The safari will have three pokémon in it if your friend has beaten the Elite Four. The pokémon in the safari will have their Hidden Ability if you and the friend have been online in Pokémon X or Y at the same time.
North of the Friend Safari, there is a pond with a fence around it. Use your Dowsing Machine to find a hidden Max Revive in the corner.