I have a character as just a single image.  I need to cut out the hands, feet, face, arms and legs and assign each to a new layer.  Which tool / tools allow me to just cut out or select specific areas and them to new layers?

Select the image then [Edit > Clone > Create Clone]. Use the Pen tool [b] to draw an outline around the portion to be moved. Select the outline and the clone then [Object > Clip > Set Clip]. Move the clipped shape wherever you want.

@Padd_CAD, thanks for the info.  When I use the pen tool and draw around a specific area it creates a grey object.  From here I can move the grey object, but it doesn't ever actually do anything related to the original image.  It is just drawing a random grey object.  What am I missing or doing wrong?

Maybe this helps illustrate my proposal.