Get in the water outside the scientist’s lair to head East onto land, and then go South a few steps over a wood bridge so you can hop through some water going East.  This lands you near a small room where you can find a Nugget hidden in the floor cobbling.

Go West into the water again and cross West onto land so you can once more head South.  Run South as far as your feet can take you, then turn East and hop into the water from the wood plank bridge.  Use that to head North and then East, popping out onto the wood bridge by the pacing janitor again.  Run past him to the North and jump in the water at the first opportunity, Surfing to the South.

When you reach the metal gate that stops you, hop onto the land to the West and go grab the X Sp. Def there, a battle item that boosts Special Defense one stage and is mostly useful for selling to get a small amount of cash.  Hop back in the water you used to get here, and head North to return to where you jumped in.  Head a little further North and you’ll find the edge of the sewer- turn to go East and follow the wall to the next bridge, which returns you to the stairs and thus back out of the sewer.

Naturally, there’s no water in the desert-like area North of Castelia, Surf-able or otherwise, and the path from Nimbasa to Driftveil is just a giant bridge that doesn’t even get near the water, so this is where you Fly back to Driftveil.  Return to Route 6 and stop off in the river between the first two bridges to pick up another Heart Scale.  Keep ahold of these- they’re used by the Move Relearner to restore a move that you missed a chance at (or never would have gotten a chance at) on a pokemon of high level or completed evolution.

When you reach the Deerling research facility, cross the river to the East to grab a PP Up item from behind the grassy patch and note the presence of the cave to the North.  There’s also a sellable Big Mushroom in the empty spot at the corner of the grassy patch.  PP Up are supremely valuable and very rare items that will increase the max PP of a move by 1/5 of its original value- only three can be applied to each move, and the effect sticks with the move: learning a new move over it will remove the effect of the PP Up.  Because of this, these items are best saved until your pokemon have finished learning new moves and you’re sure what the final move sets you use will be.  The Big Mushroom can be sold for cash, or you can keep it until you find a Fan who will buy it (there’s one for rocks too), or you can bring it to the maid in that one trailer so she can use it to cook something.