You'll pick up this quest in Chapter 3, after clearing out Scrap Boulevard with Tifa. Talk to a Junk Dealer running shop behind Seventh Heaven and he'll tell you about some darkes that have moved into the abandoned Talagger Factory. Worried about his regulars, he'll send you to deal with them... or rather, to talk to somebody named Narjin, who will fill you in on the details. Before you go, note that you can buy Potions and a single Ether from this Junk Dealer at a reduced cost, should you feel the need.
Make your way east to the Pillar Plaza and head to the northwestern exit, which was previously sealed. Here you'll find Narjin standing guard, and he'll let you pass now that you have Tifa with you. Go through the gate and enter the Abandoned Factory.
|Lesser Drake||Wind||Orb of Gravity|
This might be a mere side quest, but this place is fairly large, and not a bad hunting ground if you're keen to do some grinding. Follow the path northwest then go under a partially open sliding door to reach the interior of a building. Smash some Shinra Boxes, then head northwest to find a fenced-off area housing a chest (2x Antidotes) and beyond the fence a tantalizing orb of magic materia. You can't reach it yet, but keep it in mind.
Return into the warehouse and exit to the southeast, turn northeast and smash some Shinra Boxes against a fence, then head northwest into another building. This time you're not alone, as you'll have to fend off several Gorgers, who should be cake by now. Kill them, smash some more Shinra Boxes in the corner, then exit to the north.
Follow the linear path ahead until you reach an intersection, at which point head northwest and follow the path until you reach the interior of another building, where more Gorgers await. Crush them, smash some Shinra Boxes in the northern corner, note the doorway on an elevated platform to the east, then continue south to find a mostly-sealed door. Hold [TRIANGLE] to interact with a device near the door to open it, then continue through to the south to claim the magic materia that taunted you earlier. Turns out to be another chunk of Fire Materia, which might be mundane... but it's free! There's also a bench you can rest at, some Shinra Boxes to smash and a vending machine which will dispense Potions (50G), Phoenix Downs (300G) and Antidotes (80G), should you somehow find yourself tried by this area.
Return to the building to the north and head upstairs and through the doorway to the east. Pass through the hallways beyond and into another warehouse building where you'll find two Lesser Drakes. They've got a fair amount of HP and can deal decent damage with their "Tornado" attack, but otherwise they're mostly nuisances for being aerial enemies. Hit them with Aero to make them more susceptible to staggering and deal significant damage, then go to work with melee attacks - your normal go-to attacks to inflict Stagger, like Focused Thrust, will likely just miss, so don't bother.
Once these two Lesser Drakes are dead, smash some Shinra Boxes to the east and note the locked door. Nothing you can do to get in there yet, so exit out the northern door and follow the path until you reach an intersection. If you head southwest at this intersection you'll be back at the fork that ultimately took you to the Fire Materia, so don't bother with that route and just head northwest into another building, where the final Lesser Drake resides. Again, use Aero to weaken it, then take it down with normal attacks. Decline the opportunity to automatically return to the quest-giver and instead smash some Shinra Boxes to the north, then open a door to the south. In the fenced-off area outside you'll find a chest containing 2x Sedatives.
Return to the quest-giver at your leisure, which will likely require you to kill any Gorgers that have respawned. Tell the Junk Vendor the news and he'll give you 500 Gil and tell you about "another flying bastard" that needs killed. This unlocks the quest Just Flew In from the Graveyard. You can also buy three more Hi-Potions from him at a good price, if you have a need.