After leaving Virbank City, you are now in Castelia City, one of the most populated cities in the Pokémon universe. Your rival leaves off almost immediately after docking, but not before he gives you his Xtransceiver contact. After he leaves, you are free to look around almost the entire city. When entering, however, a harlequin stops you and gives you a bicycle. Then he asks for you to find 3 other harlequins in different buildings (Medal Office, Passerby Analytics HQ, and Battle Company, to be precise.) If you find and talk to the harlequins and go back to talk to the one outside, he gives you a Rare Candy.
Going down some of the streets lead to different things. The left most one leads to the Medal Office and the Gym. If you speak to Clyde, he tells you that the Gym Leader, Burgh isn't there. After that, Iris shows up and guesses that Burgh is off getting over artist's block. She also says that Team Plasma, despite being broken up 2 years prior, may be hiding out in the sewers, and we should investigate. Clyde stays blocking the gym, and tells you where the Pokémon Center is.
The next one over is a back alley. In here, there's a cafe that contains the Tutor for Meloetta's signature move, Relic Song. Also in the back alley are two Roughnecks that don't have much to say. However, behind the dumpsters in front of the cafe is a man who will give TM 70 Flash, a move that illuminates caves and the like. This move also lowers accuracy in battle (although, he makes a mistake when saying it takes two uses to halve accuracy. It is actually three uses.)
The alley behind the harlequin contains a vendor for Casteliacones. These are like Lava Cookies or Full Heals from other games, so stocking up is an option, but they come at 200 bucks a piece, so don't shell out too much money at this point of the game. Also back here is Studio Castelia.
Down the street next to the Pokémon Center is the headquarters of Game Freak. More things with this building are unlocked in the after game, so there is not too much here at this current point in time. However, on the 22nd floor, there is a Revive on the ground. The other building on this strip also contains TM 44 Rest, a move that puts your Pokémon to sleep for 2 turns, but fully heals them of status (aside from Sleep and Freeze) and all of their HP. This is found on the 11th floor.
All the way to the right leads to Castelia Sewers, in which Iris interrupts you when making your way there to remind you where to go. Hugh makes a mad dash into said sewers, and Iris recommends you check it out.