How do I tell what needs repaired/replaced/cared for? What items on an older bike should I check for wear and damage and how do I differentiate wear/damage that can safely be ignored and wear/damage that I should repair/replace? 

At the start of this summer I started riding my dad's old bike (a 2008 Specialized Hardrock Sport) to/from my summer internship, and I expect to continue riding it regularly when I go back to classes in the fall. Until this summer it was only ridden occasionally, and spent most of the time in the garage. As far as I'm aware none of the original components have been replaced. Everything seems to run fine at the moment (after I re-tuned the derailleurs and tightened the brake cables), but many components have very visible wear-and-tear, and I don't always know how to differentiate cosmetic damage from other damage.

Ideally I'm looking for a list of common components to fail or need replaced, and how to tell if they are about to fail/are ready to be replaced.

I currently have to store the bike outside, if that changes what I should be looking for.
Clean and oil the chain regularly.  A "chain washer" (about $30) is good for cleaning while making minimal mess.

Since you have to store your bike outside, use "wet" chain lube (at least in rainy season).

Buy yourself an inexpensive "chain wear indicator" for about $10.  Replace the chain when it's stretched to about 0.8, and, when you replace it, check your cogs to see how worn they are.  (The Park Tool site likely gives some hints as to how to check for cog wear.)

Visually check your brake blocks every now and then (assuming rim brakes), and replace before the metal part comes close to touching the rim.

Occasionally inspect your cables for fraying or excessive rust, and replace if indicated.  As the bike is stored outside, occasionally put a few drops of lube on the cables where the inner cables slide into the outer housings.

Occasionally inspect your tires for wear, and for cracks in the sidewalls that indicate they are "drying out".  (And, of course, get a good "floor pump" with built-in gauge and check your tire pressure at least weekly.)