In this city:
In the Pokemon Center you'll find Mr. Bonding. Talk to him to receive PP Restoring Power LV1. To the north of the Pokemon Center, there are two houses. The one to the left has an NPC that can teach Mythical Moves to Keldeo (Water/Fighting) or Meloetta (Normal/Psychic).
Go upstairs and talk to the girl in the middle of the room to get TM08 Bulk Up. The next house only has an NPC that can change the BGM into a much soothing, piano music. However, you'll just be able to hear it when entering that house.
Head to the northwest and get the hidden Icy Rock before reaching the wooden bridge. To the northwest portion of the city, past the wooden bridges is a house that has a woman that will trade your Jigglypuff to her Bisharp. You can catch a Jigglypuff in Route 20 – Winding Woods.
From that house, continue south and you'll find an information board to the left, right after the stairs. Slide down the slope to the west then walk to the south to pick up the Full Restore there. Slide to the south and follow the stairs north to find a house.
Examine the tree outside the house to find a hidden X Sp. Atk. Enter the house and you'll find the old man also known as the Move Tutor that can teach new moves to any of the Starters, provided that they've achieved their last evolution. Fire-type starters can learn Blast Burn, Water-types can learn Hydro Cannon and Grass-types can learn Frenzy Plant.
There's a photo spot in the middle-southern portion of the city. Next, head towards the gym and find the hidden Full Heal in the west side of it, near the boutique. Finally, you can check out the boutique and buy some new clothes for your character.
When you thought you're ready for the gym leader, you'll only discover that he's gone to the Winding Woods. Make your way to the southwest corner of the map then follow the road until you reach the forest.