First, move across the bridge. Follow the path left until you reach another bridge. You now have two choices, you can either take the route south for a detour where you'll find a number of items, battle a number of trainers and have the chance to extensively train your Pokemon.
Or you can continue left, immediately taking on the Team Plasma Grunts. We assume that you want to explore the entire area, but if you don't just skip ahead to where we backtrack to this point and take the route left.
Move over the bridge and head south. The path leading west is guarded by a man who refuses to let you past. Instead, continue south until you reach a Janitor. This guy's ready for a fight!
Trainer Battle: Janitor
Once you've won, continue south. You'll reach another bridge leading west--go across it. In an alcove sits an item, which turns out to be an X Special . From there, head north and continue west behind the wall. You'll see a Clerk Male randomly standing around.
Run past him and continue along the path. You'll move over a second bridge. Go north until you reach two bridges both leading north. Take the right one first. You'll find a Worker here that's willing to battle.
Trainer Battle: Worker
Once defeated, go back over the bridge. This time, take the bridge on the left leading north. Continue north over the second bridge. Here you'll see a Doctor. Talk to him if you want to fight.
Trainer Battle: Doctor
From now on, anytime that you talk to the Doctor that you just defeated, he will heal your Pokemon . As your Pokemon already automatically heal between battles because you're travelling with Hugh, there's little need for this. But once Hugh has left, feel free to utilize his services.
Anyway, go right, past the door. You'll notice another item on the ground. Move over and pick it up. It's someone's Leftovers . Now enter the door that you just moved past.
Inside, a scientist will give you a random healing item from a small range of possible ones. You can return here once per day to receive another. Once you're received it, leave the room. There's nothing important over the bridge to your west, so instead move back over the one leading south, continuing over the second one.
Now take the bridge west, followed by the one that leads north. Collect the Rare Bone lying on the ground on this platform then move back over the last bridge.
There's nothing else to do in this part of the sewers, so backtrack all the way back, past the Janitor and the man that was guarding the staircase until you're back at the point where you chose to go south instead of west. This time, go west.
Pick the TwistedSpoon up off of the ground then move over and approach the Plasma Grunts. After speaking with them you'll enter into a Double Battle.
Double Battle: Team Plasma Grunt and Team Plasma Grunt
The Grunts will scram after you defeat them... and Burgh, the Gym Leader will appear! After talking, Hugh will give you HM04, which teaches Strength . He will soon leave. A strange man will then appear from the door to your north and promptly leave. Finally, Burgh lets you know that he's now back in his gym and also leaves.
It's now back to single battles again. Take the slightly obscured path to your left. Follow the path until you reach the item at the end. Pick it up to find that it's a Heart Scale . Backtrack to where you battled the grunts. If you enter the door to your north, you will enter an entirely separate area known as Relic Passage .
This is only part of the Relic Passage, but it isn't too big, has a few trainers and a few nice wild Pokemon, so you might consider exploring it. If you wish to explore it, do so, otherwise exit the sewers at the same place that you entered it.