I'm fairly new to Blender. I know there is the "Transform" section in the Object menu of the Properties tab, which contains Location/Rotation/Scale of a selected object. What I'd like to know is if there is way to tell the height (Z) of an object, or just the selected face?
In Blender 2.5 and later, select the face or edge you want to measure in Edit Mode, and turn on the Properties shelf by pressing 'N'. In here, scroll to Mesh Display > Numerics. You can select to display the edge length and the face area of the faces.
To get the size of an entire object, the Properties shelf in Object Mode will list the X, Y and Z dimensions. It's right in the Transform section.