Welcome to the Pokemon World Tournament or PWT for short. If you go along the west part of this area, you can find an entrance to the rest of Relic Passage , which you can also use to return to Castelia City . If you want to enter that area, feel free to skip ahead to that section.
First up, move south and turn your Dowsing MCHN at the corner of the path. A hidden item will show up on it, stating that it's to your south. Follow the MCHN's directions to find the hidden Ether . Move back to the path and head east. Along the path you'll notice three stalls.
The one north of the path will tell you a selected Pokemon's Hidden Power type . The left stall owner south of the path will allow any of your Pokemon to relearn old moves which they've forgotten to replace new ones and the stall owner next to it will allow you to make your Pokemon forget any of their moves .
Feel free to use these stalls at your leisure. Once you're ready, continue east to the Pokemon World Tournament building itself. You'll come upon Hugh and Clay standing outside the building. They'll talk for a bit then you will all head in.
Once you're inside Clay will talk with a trainer planning to enter the Driftveil Tournament . He'll head in and Hugh will eagerly follow him. You'll be free to enter too if you wish and will eventually need to, to advance the plot.
First you may wish to exit and grab a few items. You'll find yourself at an intersection. Take the path south first. You'll head onto the docks. When you reach a T-intersection, head west. At the end of this dock is a PP Up! Grab it and head back to the main intersection in front of the Pokemon World Tournament building.
This time, take the path east. Along the this path are two more stalls. The east-most one has a Move Tutor that will teach Frenzy Plant, Hydro Cannon or Blast Burn to a starter Pokemon from any generation. Similarly, the west-most one teaches Fire, Water or Grass Pledge.
At the very end of the path is a man looking north. Talk to him to receive a Rocky Helmet . From there, head north until you're past the fountain. Move west and you'll see two Vending Machines with a bin to either side. Look in the eastern bin to find some Zinc . That's all there is to do here!
If you want to, you can check out the Relic Passage before you enter the Driftveil Tournament in the Pokemon World Tournament building.