After beating the Champion you’ll start the game back in your room in Vaniville Town. Sweet! Head downstairs and speak with your mum. She’ll mention that Sycamore wants to meet you at Lumiose Station . Better not keep the Prof waiting then. Outside, Shauna will grab you as you leave. She’ll mention that she wants to trade.
This is great; she wants to give you a Fennekin if Froakie was your starter, a Chespin if Fennekin was your starter or a Froakie if Chespin was your starter. Give her any Pokémon you have in your party or stored in your PC. Once you’ve traded, Fly to the north side of Lumiose City . You should be in North Boulevard .
Head clockwise from the Pokémon Center and you’ll eventually see Lumiose Station to your left. Don’t be shy; head on in!
Sycamore’s assistants will appear when you enter the station. They’ll upgrade your Pokedex to add the National Pokedex function. Now you can view every single Pokemon ever! After pocketing your shiny new dex, head north-east over to Professor Sycamore. He’ll give you the TMV Pass . Now you’ll be able to visit Kiloude City!
Whenever you want to head there scan your pass on the orange machines. If you want to head there now, go ahead! Kiloude City houses many new attractions, not least the Friend Safari , where you can find rare Pokemon and all that! If you don’t want to head there, well there’s a few other things to do too!
Now that you’re champ, three additional legendaries await your immediate capture: Zygarde and Mewtwo found in Terminus Cave and Unknown Dungeon respectively, plus one of three legendary birds roaming around Kalos . You can do anything else you want to too, such as battling with friends and the like. Just enjoy yourself!