When you first arrive in Driftveil, Clay will express annoyance with you and Cheren, because lowering the drawbridge allowed some Team Plasma people to get into the city. He says that he'll keep the Gym closed until you and Cheren round up Team Plasma.
Exploring the Town
Now you can explore the town. There is a person in a house who will give you a Shell Bell. Also, in the Pokemon Center, a girl upstairs will give you three repeat balls. Also, there is a large market where you can buy things like herbs. Someone in the market will give you a Black Belt if you show him a pokťmon that is level 30 or greater. If you go into the market by a door on the left side of the building, you can get a Big Pearl.
There are number of hidden items as well, so be sure to turn on your Dowsing Machine.
Finding Team Plasma
Team Plasma is hiding somewhere in Driftveil City. Go south and check out Cold Storage.