After you get the Poké Flute and get the Snorlax off of the bridge, you can visit the west side of Rivière Walk.
Pokémon Day Care
West of the bridge, you find the Pokémon Day Care. Trevor and Tierno will ask you to go in and check it out with them. You can leave up to two pokémon in the day care. If they are compatible, the Day Care might find an Egg after a while. If the day care has found an egg, the day care man outside of the Day Care will be facing away from the Day Care instead of watching the pokémon in the fence. If you get an Egg and keep it in your party while walking around, it will eventually hatch into a pokémon. That pokémon might even inherit moves from its parents that it would not otherwise be able to learn.
Pokémon Breeding
To cause an egg to appear, you have to leave both a male and a female pokémon at the day care, unless one of the pokémon is a Ditto, which you can only catch much later in the game.
If you leave a male and female pokémon at the day care, they have to be compatible in order to produce an egg. You can talk to the guy outside of the day care to find out if they are compatible. If he says that they "would prefer to play with other pokémon," it means that they are NOT compatible. Any other message from him means that they are compatible.
Ditto is compatible with most other pokémon: if you leave a Ditto with another pokémon in the day care, in most cases they will produce an egg, and the egg will hatch into whichever pokémon type you left in the daycare with Ditto. (You can't get a Ditto from an egg.)
If you don't put a Ditto in the day care, then the pokémon that hatches from the resulting egg will be the same as the female parent.
Some pokémon, such as legendary pokémon and baby pokémon, are not able to produce eggs. For more information about pokémon breeding compatiblity, please check out Bulbapedia's Egg Groups page.
Battle and Catch
Go south from the Day Care to find Tierno in the flowers. He asks if you've seen Dunsparce.
In the flowers and grass, you can find Croagunk, Smeargle, Volbeat, Illumise, Roselia, Ducklett, Spritzee (Y only), Swirlix (X only), and Flabébé (orange-flower and white-flower). In the yellow flowers, you can find yellow-flower Flabébé, and in the purple flowers you can find blue-flower Flabébé. You might also have horde encounters with Psyduck, Hoppip, and Roselia.
To the west, you find Artist Georgia. She has a level 16 Smeargle.
Trevor is the flowers to the west. He'll ask to compare pokédexes with you to see who has seen more. South of him, there is an X Sp. Def.
West of that, you encounter Artist Family Mona & Paolo. They send out two level 16 Smeargles.
Battle Chateau
To the west of the Artist Family, Trevor shows you the Battle Chateau. Inside, Viola is there talking to Hennessey. In the Battle Chateau, you can be granted titles and join the nobility. Viola vouches for you, so Hennessey gives you a starting title of Baroness. If you battle in the Battle Chateau, you can increase your nobility rank. With a higher rank, you can find higher-ranking members of the nobility in the Battle Chateau to battle against.
The nobility in the Chateau are random and change throughout the day, making the Battle Chateau a good place to earn money and experience.
Other nobility may come in as you go around the Chateau. There are four areas where you can find nobility to battle: a room northwest of the lobby, a room northeast of the lobby, the hallway to the left and right of the lobby, and a back room accessible from the hallway.
If your noble rank increases, the maid will tell you your new title when you leave.
Go West
West of the Battle Chateau, you encounter Artist Pierre. He has a level 16 Smeargle.
If you go farther west, your neighbor asks you to team up for a double battle against Trevor and Tierno. In the battle, you fight Trevor's level 14 Pikachu and Tierno's level 16 Corphish. Trevor also has a level 15 Flabébé.
South of where you battled Tierno and Trevor, you can get a PP Up.
To the west, there is a Photo Spot. West of there, you find the entrance to Connecting Cave, which provides a shortcut to Cyllage City. You can enter this northern cave if you wish, but the path to Cyllage City is blocked. Back outside, go southwest to go toward Ambrette Town, where your neighbor said you should go next.
After you go up two flights of stairs, go north. If you have Cut, use it to cut the prickly tree, then follow the path to find Silver Powder. Go south from there and through the tall grass, going west through the grass to find a Persim Berry tree. South of the tree, there is a Tiny Mushroom.
Go back the way that you came and go into the southwest cave to enter Connecting Cave.