The shopping area of Sector 5 slums?

First down then up then straight ahead, that's just what I say,
The woman who first goes around looking down for things on the ground, later looking up for meteor, and later yet looking straight ahead—or so she says, but her sprite looks the same.

Then you'll find a best friend - hey get out of the way!
Might be a dog there? Not sure.

Believe in oneself, while looking at the screen,
Dunno about this one.

Go up but don't take it - that would just be mean!
What seiferalmasy2 said.

Scrap metal and materia, you can find it here,
This guy needs an antidote, and you a brain, my dear.
Same as I said before.