After you become Champion and enter the Kalos Hall of Fame, there are more things to do in the post-game.
After you go to the Hall of Fame to record the team that helped you become the Champion, some scenes will take place, then AZ will ask you to battle him. He has a level 60 Torkoal, a level 60 Sigilyph, and a level 60 Golurk.
The post-game begins the next time you start the game after you have watched the credits roll. Here is a list of new things you can do in the postgame. This list is just a summary; there are more details in the next section.
After you become Champion, battle AZ, and watch the end credits, then the next time you start up the game, you'll start in your house in Vaniville. Your Mom will say that Professor Sycamore wants to meet you at Lumiose Station.
When you go outside, Shauna is there. She wants to trade pokémon with you. She'll offer you the starter that she originally chose (it's the starter that is weak to the one you originally picked). You can give her any pokémon you wish. You'll be able to choose from the PC Box. Shauna's pokémon is level 5.
If you don't have a pokémon with you that knows Fly, you'll either have to teach it to one of your Hall of Fame pokémon, or you'll just have to walk/rollerskate/bike to Santalune City to reach the Pokémon Center. After you have a pokémon with you that knows Fly, you should fly to northern Lumiose City.
Once you are in Lumiose City, go to Lumiose Station. It's in North Boulevard. There are a couple of red-skirted girls standing in front of it. It's a light-brown brick building with an arched door.
Inside Sina and Dexio will arrive and give you the National Pokédex.
After that, go talk to Professor Sycamore. He gives you a TMV Pass so you can ride the train to Kiloude City, which is in southern Kalos. After Professor Sycamore departs, check on the orange machine to ride the TMV to Kiloude City.
Go to 2F of Professor Sycamore's Lab in Lumiose City. Talk to the scientist on the right side. He'll give you the Poké Radar. You can use the Poké Radar while standing in grass, flowers, or water. When you use it, patches of grass/flowers will shake or water will bubble. Go to one of the shaking spots to encounter a pokémon. Defeat that pokémon, then the Poké Radar will show more shaking areas. Go to another shaking spot to encounter another pokémon.
The Poké Radar will keep showing shaking spots after battles, but only if you are lucky enough to keep encountering the same pokémon each time. The chain will be broken if you encounter a different pokémon, and it will also be broken if you put on your rollerskates.
You can use the Poké Radar to increase your chance of finding Shiny pokémon. The longer the chain you get using the Poké Radar, the higher the chance of encountering a shiny.
Each day, the scientist who gave you the Poké Radar will tell you which pokémon he wants you to research. Use the Poké Radar in an area where that pokémon can be found, and try to get a chain of that type of pokémon (defeat it, move to another patch of grass, repeat). Come back and talk to the scientist to get rewards for your research.
You can unlock the Looker Sidequest by upgrading the Mega Ring. To do that, you have to go to Kiloude City, win at least one battle in the Battle Maison, then go to the north part of Kiloude City and battle your neighbor. After that, meet Professor Sycamore in front of the sundial.
After that, the next time you go to Lumiose City, you should get a Holo Clip on the Holo Caster. If not, leave town and go back in and walk around. The Holo Clip is from Looker, who asks you to go to the Looker Bureau. It's near Rouge Plaza and can be found in the building across from Sushi High Roller.