Cedric will greet you and tell you about Dragonspiral Tower. Once he's done explaining, he will head off to the tower. The Pokemon Center is up the stairs to the north. The old lady inside will give you TM 31 Brick Break. It can remove the effects from moves like Reflect before doing damage. The old man by the counter will buy the Gems that you find in caves.
The Icirrus City Gym is in the northwest corner of the city. South of the gym is the Pokemon Fan Club. Go inside and talk to the old guy wearing a hat. If you show him a Pokemon that has gained between 25 to 49 levels since you caught it, he will give you an Exp. Share. He will give you a Cleanse Tag if you show him a Pokemon that has gained 50 to 98 levels. He will give you a King's Rock if you show him a Pokemon that has gained 99 levels. This means you have a level 100 Pokemon that hatched from an egg at level 1.
The house east of the gym is home to Aha, Wye, and Ditoh. They will give you a "Pep Quiz" once per day. The prize for answering the question correctly is an Antidote. You'll still get a Parlyz Heal if you answer incorrectly.
There is a house in the southeast corner of the city that is only accessible in the winter. Talk to the purple-haired woman inside to get a RageCandyBar. You can use it to battle a Darmanitan with the Zen Mode ability in the Desert Resort.
The only thing left to do in town is take on the gym in the northwest corner of the city.