Say I have a set of objects parented to another object, for the sake of this example say they are parented to an empty. If I have the empty selected, is there a way of quickly selecting all the objects parented to it?
Select the parent object, then Shift G > Children.
Note : To keep the parent selected, immediately after calling the operator, press F9 or click on the foldout in the lower-left of the 3D viewport and check "Extend". This setting will be remembered the next time you call the operator, until you close Blender.
Or, if you want to select all children at the same level, you can first select one child, then choose Slibings in the same menu.
Go to Outliner, Ctrl LMB on the parent object icon, which will select all children and itself, then Shift LMB on it again to deselect the parent.
First Select parent, then press ] to select object in the next level. Shift + ] one or more times will select more levels.