Seeking introductory texts about coordinate system projections?

One of the best intro texts I've seen is one previously put out by ESRI >

Understanding Map Projections

PDF can be downloaded here

It not only gives a good overview of projections, coordinate systems, Datums, transformations, etc: it also lists all of the projections supported in ArcGIS (8) and their best uses.

IMO, John Snyder is the "Uncle" of Map Projections. Page 5 from this pdf excerpt from his "Map Projections - A Working Manual" might be worth reading.

In your example of calculating area, the answer is yes, you can do it in projected space---if the projection preserves area (i.e. is equal area). For an overview of projections and their properties, see:

http://egsc.usgs.gov/isb/pubs/MapProjections/projections.html