After exiting from Pinwheel Forest, you are on a little plaza from which you can see Skyarrow Bridge. Go into the toll house and talk to the guy with the backpack and you will get a Quick Claw. Then you can go through the other exit of the toll house and you will find yourself on Skyarrow Bridge. Enjoy the view as you walk across this huge bridge. There are a few people to talk to here, but no trainers.
Finally, after crossing the bridge, you will reach Castelia City, which is a huge city full of skyscrapers and tons of people.
There are a great many places to check out here. On the main street that you start on, you can find the Pokémon Center, the Battle Company, and the Passerby Analytics HQ.
Battle Company
At the Battle Company, you will encounter a bunch of trainers on the upper floors. On 47F you will encounter trainers who have Normal, Fighting, Rock, Grass, and Water type pokémon. The moves that these pokémon have will include Normal, Dark, Fighting, Rock, Ground, and Water. The highest level pokémon on this floor is level 19.
On 55F of Battle Company, you will encounter Psychic/Flying, Psychic, Bug/Grass, Fire, Grass, Water, Poison, and Normal pokémon. These pokémon will have moves that include Psychic, Dark, Flying, Normal, Bug, Grass, Ghost, Fire, Water, and Poison. The highest level pokémon on this floor are level 20.
Passerby Analytics HQ
Passerby Analytics HQ is a place where they take surveys. One person will ask you your job and hobby. When you answer, these items get added to your trainer card. Another person there will have a bunch of different surveys for you. If you take all of the surveys, they will all be automatically atted to your C-Gear survey module.
From the main street, you can reach Liberty Pier, which is where you can depart for Liberty Island if you received a Liberty Pass in the download event. On Liberty Island, you can catch Victini. Read more about catching Victini on Liberty Island. There are a number of other piers that you can reach. One of them has the Royal Unova cruise ship.
There are also some streets leading north. On one of these streets is Game Freak, where you can talk to a girl in the lobby to see a picture of Zorua, which adds a Zorua sighting to your Pokédex. Another street has an art gallery, where if you bring a pok&ecaute;mon of the type that the artist requests, he will give you a berry of your choice. You can only do this once per day.
There is a street with a Casteliacone ice cream stand. The stand is closed during Winter, but in all other seasons you can buy a cone on Tuesdays. You can only buy one cone per week. It can be used to cure all status ailments of a pokémon.
Be sure to look for the many vending machines around the city. You can get Fresh Water, Soda Pop, and Lemonade. These items heal HP and are much less expensive than potions that cure the same amount of HP. It can be kind of tedious to buy a large amount of items from the vending machines, but it is certainly worth it.
There are three Dancers around town. Talk to the dancer at the fountain in the area north of the side streets and he will say that he wants to start a team of dancers. If you battle each of the dancers in town, they will all go to the fountain and you will get an Amulet Coin from them.
On Narrow Street, a guy will pop out from behind some trash cans and give you the TM for Flash. One of the dancers is on this street. Also, in the café on this street, the bartender will give you a free Lemonade.
Where is Team Plasma?
If you head into Burgh's gym, Cheren will step out and say that he defeated Burgh. After Cheren leaves, Burgh himself will pop out and say that he can't challenge you right now because Team Plasma has shown up. He asks you to meet him at Prime Pier.
Where is Prime Pier?
Prime Pier is the one that you can reach if you stand between the Pokémon Center and the Battle Company, then go south from that intersection.
At the pier, you will find Bianca and Iris. Team Plasma stole Bianca's Munna! Burgh says that since Castelia is such a big place, it might be tough to find Team Plasma in the crowds. He will leave to look around town. Go find him at the fountain in the north part of town, near the western road. He will go down to that road, so follow him. You will find Team Plasma standing outside of a building near Burgh's gym. You must fight a Team Plasma Grunt who has two Sandiles.
When you go into the building that the Team Plasma guys went into, and you'll find three of the Seven Sages: Gorm, Ghetsis, and Bronius. They will talk about the legend of the formation of the Unova region, in which a hero found a legendary Dragon pokémon (Reshiram or Zekrom, depending on which version you're playing) and used the power of that pokémon to bring unity to all people. Team Plasma seems to think that they can use this legendary pokémon to win the hearts and minds of people not to create peace, but so that they can remake the world as they see fit.
Burgh leaves and goes to his Gym, so your next step is to challenge his gym.