After you explore Castelia City, you find out that Burgh isn't in the gym right now. Iris tells you to look for Team Plasma in the sewer. Hugh goes to investigate, and you should, too.
You and Hugh will be doing double battles as you go through the sewer. He sends out his starter, which is at level 20 right now. He will heal you up after every battle, so you shouldn't need to use any healing items.
The layout of the sewer will be different depending on the season. The water will go down in August.
If there is water, go north up the wooden bridge. If there is no water, go down the stairs into the lower area.
Head west. You will see some Team Plasma Grunts up above, but first, keep going to the left to get a TwistedSpoon.
Now go north and talk to the Team Plasma Grunts to fight them in a double battle along with Hugh. They have a level 16 Scraggy and level 16 Sandile. When you defeat them, they run away.
Burgh arrives and says that he didn't find any other suspicious people, so there's no need to explore farther.
Hugh gives you HM04 Strength to thank you for helping, then he leaves.
After that, a strange, blonde, glasses-wearing guy with a weird blue hair thing walks by and compliments you on the battle.
Burgh says that you can explore the sewer some more to train your pokémon, or you can go challenge Burgh at his gym.
If you go south and down the wooden bridge, then go left, you find stairs leading up to Castelia Garden. You can catch some wild pokémon here, including Eevees. There is also a Miracle Seed here.
Back in the sewer, you can go south and challenge Janitor Felix, who has a level 16 Lillipup and a level 16 Trubbish.
South of him, there is an X Special.
When you leave the sewer, Iris says you can challenge Burgh now.