To start with, a woman can't get into her apartment because a crate is blocking the doorway. You will need a Makuhita to push the block away. And when she gets into her apartment, there is a Venonat for you to deal with. After you capture it, head out and a woman named Teresa wants you to get her five Skitty back. Their locations are listed here:
A Skitty is on the roof of the apartment you were just in.
A Skitty can be found at the fountain which is at the south of Fall City.
A Skitty is behind a wooden fence which can be cut down by a Meowth.
A Skitty is in a house at the north-west of Fall City, disguised as a plush doll.
A Skitty is in the right-hand trashcan near the base. Use either Makuhita or Snubbull to open it.
After you have given Teresa her five Skitty back, you will meet Prof. Hastings' assistant who needs her research lab basement lit up. At the very south of Fall City (the harbor), the Snorlax has moved, allowing you to go down the ladder that was underneath it. Climb down and capture the Staryu and use its field move in the research lab basement to light it up. Head out, and you will find out a Machoke is wrecking the Clock Tower. Capture it and it's Mission Clear!