What class would this be covered in? Would the material in this section of a wikipedia article be covered in a standard course on Differential Geometry, or should I look elsewhere to learn those sorts of things? Specifically, topics like solid angles, line/surface/volume elements, and so forth. I have taken vector calculus, which did not get into these topics in any detail. 

A standard differential geometry class would certainly contain this material, but it would probably not spend significant time on them. This is more the flavor of a multivariable calculus/analysis class; if this wasn't covered to your satisfaction in yours, you might want to study these on your own. 
