When you arrive in Mistralton City and walk a little bit to the north, Cedric Juniper and Skyla arrive. Cedric is Professor Juniper's father. He will upgrade your Pokédex to make it show you all of the alternate forms of pokémon. For example, male Unfezants look very different from female Unfezants. The upgraded pokédex will let you see that, and other pokémon that have different forms.
Skyla says that she needs to go to Celestial Tower, where she saw a sick pokémon when she was flying over. Celestial Tower is a place north of town that you can reach by going up Route 7. You'll have to go find her at Celestial Tower before you can challenge her Gym.
In the Pokémon Center here in Mistralton, there is a guy near the Nurse who can check on one of your pokémon and tell you what its Hidden Power would be. Hidden Power is a move that a pokémon can learn from TM10, which Professor Juniper will give you after you have seen a certain number of pokémon. The damage type that Hidden Power inflicts (for example Fighting, Dark, Water, etc.) depends on the pokémon's IVs, which are hidden values that each pokémon has. The guy in the Mistralton Pokémon Center will tell you what type this would be for the pokémon that you show to him.
An important location in Mistralton City is the house of the Move relearner and Move deleter. The Move Relearner will teach a pokémon any move that it is at a high enough level to learn, in exchange for a Heart Scale. To find Heart Scales, you can talk to a woman in Driftveil City who asks to see a pokémon that knows a specific move. She will give you a Heart Scale in return. Otherwise, you can do the Feeling Check with the C-Gear to get Heart Scales.
The move deleter will make a pokémon forget a move of your choice. This is the only way to get a pok&ecaute;mon to forget an HM move. The move deleter will do this for free.
Check all around the airport for items. There is TM58 Sky Drop near the plane at the southern end of the airport, and there are a bunch of hidden items around the greenhouses at the southern part of the airport. Also, inside of the Cargo Service building, a woman will give you a Sharp Beak.
Once you've checked out Mistralton City, go north to Route 7.