After encountering some Team Flare Grunts in the stone field of Menhir Trail, you reach Geosenge Town.
In the Pokémon Center of Geosenge Town, talk to the guy in white to get TM66 Payback.
There is a hotel to the right of the Pokémon Center. The girl in the middle room upstairs gives you a Downcast Ribbon. Mr. Bonding is in the room on the right. He gives you a new O-Power: Speed Power.
Behind the hotel, there is a Timer Ball in the fenced-in area.
When you go north of the Pokémon Center, you see a Team Flare Grunt singing about Geosenge Town. The Grunt then runs to the left.
In the southwest corner of town, there is a Soft Sand.
North of there, there is a cabin. Inside, a researcher gives you an Everstone. A pokémon holding this item will not evolve.
To the east, there is a Photo Spot where you can call Phil the Photo Guy to take a photo of you in front of the monument to the north.
East of there, a Pokémon Ranger is telling a few backpackers about the monument. The backpackers are blocking the path to the east.
In the northwest of town, you find the Team Flare Grunt that you saw earlier. The Grunt says that they are going to use the stones on Route 10, and a fantastic treasure, to bring Team Flare happiness. Then the Grunt runs northwest. Afterward, your neighbor runs up and says that the way that the Grunt went is a dead end.
Go that way and you will discover that there is a big stone structure, but the Team Flare Grunt is nowhere to be found. The stone structure looks pretty suspicious to me.
Go south again. Near the chairs and tables outside, you find the guide who was talking to the backpackers earlier. She says to go to the hotel to learn more about the artifacts. In the hotel, she is standing near the stairs. She talks about the stones, then says that Shalour City, which is next to Geosenge, also has some mysterious stones.
Go outside and go to the exit on the east side of town that the backpackers were blocking earlier. Leader Korrina will come up and say that her Lucario wants to battle you. She wonders if you will be able to tell which Lucario is the one that likes you. The battle begins right away. She has a level 25 Lucario, followed by another level 25 Lucario.
After the battle, Korrina says that she's the Gym Leader of Shalour City, and you'll get there by going east and going through the cave, so you should challenge her when you get there. Go east from here to reach Route 11: Miroir Way.