After kicking Team Flare out of the Kalos Power Plant, the power is restored to northern Lumiose City.
When you enter the city, Shauna runs up to you and asks you to go to Prism Tower with her. She won't let you explore the city until you have gone to the tower together, so go there now. You'll find Clemont, the Lumiose City Gym Leader. He turns on the lights.
Navigate the Town
Lumiose City can be very confusing. Its circular design makes it hard to figure out where you are or remember where anything is. An easy way to find what you're looking for is to talk to a Lumi Cab driver and choose the location that you want to go to, but don't pay to be taken there. The cab driver tells you which street the location is on. After you know which street you need, then if it's not North or South Boulevard, go to Centrico Plaza (where Prism Tower is) and read the street signs that are on the sidewalks of each big street. That way you can quickly find the street that you're looking for and go to the location that you need.
Become More Stylish
Now that you can visit the northern half of Lumiose City, there are more things you can do to become stylish. The full list is available in the Lumiose City Style Guide.
The Hotel Richissme is near Café Ultimo, across from Hibernal Avenue. Mr. Bonding is in the hotel lobby. Talk to him to get Bargain Power Lv. 1 from him.
Use the elevator to go to the top floor of the hotel, 5F. On the right side, there is a girl standing in a bathroom. Talk to her to get TM49 Echoed Voice.
Talk to the girl at the counter to get a part-time job. You can do Room Service, Making Beds, or Lost and Found.
You will get a phone call with a food order. You need to remember the phrase that the person said. The phrase goes like this: "A [Nature] [Berry] [food type], [Move] style." For example, "A Careful Aspear Berry Canapés, Knock Off style." After the phone call, the guy behind the counter will ask you to repeat the order.
You are asked to make all of the beds on the floor within a time limit. Just run to each room and check on the bed. The minigame ends as soon as you finish making the last bed on the floor.
Lost and Found
You have to find an item that somebody lost in the hotel room, but don't step on it! To find the item, check in front if you, then take one step, check in front of you again, and so on. Make sure to check the spot in front of you before you step there, to make sure that you don't step on a hidden item. The game ends as soon as you have found all of the missing items.
In the Pokéball Emporium, talk to the girl in the middle and say you like round things to get some Heal Balls or Luxury Balls.
The Pokéball Emporium is is a great place to increase your style. You should first go to the Hotel Richissme and get Bargain Power from Mr. Bonding (see the previous section for more information). Then use Bargain Power and purchase one Premier Ball, then talk to the shopkeeper again and buy another Premier Ball, then repeat that process until your style has increased by the desired amount. Don't buy multiple Premier Balls at once: the number of items you purchase doesn't change how much your style increases, but your style increases each separate time you talk to the shopkeeper. That's why you should buy one item at a time.
A purple-haired guy in the Lumiose Museum on the second floor will give you TM82 Dragon Tail.
Black Office Building
Next to the Gallette Stand in North Boulevard, there is a black office building. Go inside and take the elevator. If you go to 2F, you don't get any items, but you do get to see something weird. Then go up to 3F and talk to the girl in the white shirt on the right side to get an Expert Belt. Nearby, there is a Battle Girl who gives you a Protein.
In North Boulevard, across from the Café Ultimo (near a skater girl standing by the trees), there is an office building where you can get a Star Piece from the researcher in the lobby if you have a pokémon that has the Aroma Veil, Flower Veil, or Sweet Veil ability.
The second floor of this office building is supposedly a café. Go to the fourth floor and talk to the little girl to get a Prism Scale. If you have a Feebas (which you can only legitimately get by transferring it from a Generation V game), you can give it the Prism Scale to cause it to evolve into Milotic when it levels up.
You can't do anything interesting at Café Lysandre right now, but you might want to just take a look. It's in the alley between Magenta Plaza and Autumnal Avenue.
Explore Some More
There is more to see and do in Lumiose City, such as battles in the alleyways, restaurant battles, and so on.
When you're ready, go into Prism Tower to enter the Lumiose City Gym.