Using QGIS, if I load a polygon layer and make a new column, how do I fill that column with the area of each polygon and another column with the length of the perimeter of each polygon?
Open the Field Calculator, select the new column and type the following expression
Make sure your layer is in a projected coordinate system (not lat/long) and in correct units (i.e. if your layer's projection has 'meters' as units, your area will be square meters).
Note that reprojecting layers on-the-fly does not change the units for area/perimeter calculation, so you have to first reproject using Save as ... with the correct target CRS.
You also can use fTools: Vector menu -> Geometry Tools -> Export/Add geometry columns
It updates your layer automatically by adding