Zones are defined in different ways and they need to be updated. For the first part, you can define a zone by manually drawing it, or clicking an enclosed space and measuring wither to the wall inner face or reference line. (3 ways total). For what you are doing I recommend using the manual method. After a zone has been changed, either the elements in contact with the zone are changed or moved or the zone itself is changed in shape or size, you will need to update the zone. Go to Design/Updates Zones and either update selected zones or all zones. The reason for this is that zones cause ArchiCAD to calculate multiple data points and if ArchiCAD re-calculated everytime you panned, zoomed or changed views you might be waiting a long time for the program to respond.
If area is all you need, the zone tool might be overkill. Although I prefer to use the zone tool where I can, sometimes getting an area using the fill tool is best. Activate the fill tool. In the Info Box or Fill settings dialog, place a tick mark in the Show Area Text check box. Draw your fill, when the hammer appears place the Area Text Box.