Foxtrapper, first: don't sweat this. It WILL be OK. Seriously. You probably already have all the grammar knowledge you'll ever need, you just have to know how to talk about it.
Second, your problem with "fast" is that it looks the same as an adjective (describing a thing, as in "fast Miata") and an adverb (describing an action, as in "she learns fast.")
Suppose we went back to your examples and used a word that helpfully changes its form depending on what job it's doing.
"He runs quickly." Yup, "quickly" is an adverb describing the verb "runs." (Lots of adverbs are nice enough to end in "–ly" so we can spot them.) You can say a lot of things to describe the running – quickly, uphill, away from the cops, our old friend "fast" – but all of them describe the verb. We still don't know anything about this guy "he."
"He's quick." OK, now we're talking about the guy, not the action, so "quick" is an adjective describing the person "he." Taken by itself, this sentence doesn't tell us anything about what the guy is doing. He could be quick at running, math, dumping girlfriends, or cleaning out your fridge. There aren't any verbs (besides the shortened form of "is") to let us know.
"He's a quick runner." "Quick" the adjective is back, describing the person "runner." What kind of runner is he? Fast. He could also be slow, sneaky, or experienced; those would all be adjectives too.
Summary: A noun is a person, a place, or a thing (such as Tim, Florida, or shoulder.) Adjectives describe (the Grammar Word is "modify") nouns. A verb is an action word (such as run or explode). Adverbs modify verbs.
(If you need to modify an adjective, that's another place for an adverb, as in "My sandwich was slightly stale," but never mind that now.)
I've got heaps of this stuff, so keep asking!