Giù corresponds to down and laggiù to down there. Laggiù is used if the place is far away from the speaker. Like down there laggiù can be used to refer to places which are very far away, in this case the meaning of giù / down gets completely lost. If two people are on the phone talking with each other the one who is in Berlin / Germany can say to the one who is in La Paz / Bolivie:
Che ora è laggiù?
What's the time down there?
From a logical point of view that doesn't make any sense. La Paz is 3600 m over sea level, Berlin is at 115. It would be more correct to say What' s the time up there?.
If the conversation refers to the place where the speaker is at the moment of speaking qui giù is used.
Volevo parlarti di alcune cose che succedono qui giù.
I wanted to speak with you about some things which happens down here.
There is no difference in Italian between movement and residence at a place.
Io vado laggiù per lavoro. I go there for professional reasons.