Head north. You'll see a person nearby, out on the grass. Speak with him to get an Oran Berry . Head to the north-west corner of the area to find an Ether . The south-east corner houses the first working Photo Spot in the game. You can call Phil the Photo Guy to come and take a photo of you here if you wish.
Ask about the camera to hear about how you take photos. When you're ready, take a photo! It'll get saved to your game. You can then tip the Photo Guy if you're feeling generous. If not, don't bother. Once you're done; head over to the palace gates.
You'll find out you need to pay $1,000 to enter, which is a bit crazy, but oh well. Do so and head on through. Head up to the palace doors and enter. Upon entering, you'll find out that the man in the foyer is missing his Furfrou . Time to go find it I guess.
This place is big, but there's not much to see. You can explore if you want, but the only things of consequence are as follows. Taking the hallway west and entering the first door you see, you'll find a bed that you can sleep in to recover your Pokémon's health.
Head back into the hallway and go back east. Keep going until you see a person standing by a statue. Talking with them, you'll find out they're a minstrel. They recite a small passage and you can tip them afterwards. Besides that there's nothing to do except to go back west to the main door you just passed and enter it. You'll come out by the courtyard.
Once you make it over the bridge, Shauna will stop you once more and say she'll check around the place. There's a fair bit to explore here. Move up to the statue just ahead. Now, let's explore the maze to the west of it first. Entering it from the southern part you'll find a Gardener.
Talking with her reveals that you can find HM01 - Cut here and because you already have the Bug Badge, you can use it to cut down trees as soon as you get it. Nice! Head north and you'll see a yellow Poké Ball item. Grab it and guess what? It's HM01 ! Yea yea! You can now teach it to a Pokémon and use it immediately!
Anyway, continuing north, you'll find another item to your left. Grab it to find that it's a Super Potion . Now head out of this maze through the north exit. Head into the next maze to your north. At the entrance you'll find that Furfrou you were looking for. When you approach it, it will run off.
Looks like you need to work with Shauna to corner it in the dead in the center of the maze. Although Furfrou ran north-east, place Shauna in the square you're currently in, then approach it from the south. It should run north. Now place her in the square north-west of the one she's currently in. Now follow Furfrou from the eastern side and around.
You'll chase it into the dead end. The owner will come and pick it up and tell you to meet him at the balcony when you're ready. For now; exit this maze back from the south. Now let's head east until you're between the two eastern mazes.
The one north of you has nothing of interest, the one south of you contains an item though, so head in and grab it to find that it's an Antidote . And that's it. Nothing else to do here, so head back into the palace itself.
Did You Know?
At the start of pretty much any of the Pokemon games you'll find Nintendo's latest console on the floor of your bedroom! In X & Y this is a Wii U (the NES, Super NES, GameCube and Wii have also been featured).
So let's go to the balcony. Head east or west along the main hallway until you reach the stairs at the end. Go up them, then head along the main hallway on the second floor until you enter the large flashy hallway. You'll see a bunch of arches leading outside. Go through one and you'll be at the balcony. Go up and talk with Shauna.
When you do the fireworks will start and things will get a little weird if you're a guy. Afterwards the owner of the palace will approach and ask if you're satisfied. Now ask about the Poké Flute . You'll soon get given it, followed by TM17 - Protect. The butler will then leave and you'll be alone with Shauna.
Now it's time to leave the palace and get back to the Snorlax so that you can wake it up.