The common modifiers good and well are easily (and frequently) confused.
Good is usually an adjective (a good book, a good job). Good can also function as a noun (the common good).
Well is usually an adverb (runs well, a well-written essay).
In formal speech and writing, the adjective good generally follows linking verbs such as be, seem, taste, and appear. See the usage notes below.
The redundant expression (all) well and good means acceptable.
It’s often used before a statement that qualifies or contradicts whatever it is that’s considered “all well and good.”
“There was never a good war or a bad peace.” (Benjamin Franklin)
Experiment with recipes until you find what tastes good to you.
The student officers displayed a remarkably good knowledge of the drill regulations.
Coffee thrives remarkably well in Fiji.
The students were asked to compose a well-organized essay in 30 minutes.
“On the cover, a well-adjusted-looking boy, hair stiff with hairspray, overalls starched, sat in a chair and puzzled over a Rubik’s Cube. He wore sensible shoes and an expression that said: This is all well and good, but my real passion is long division.”
(David James Poissant, “Refund.” The Heaven of Animals, 2014)
Of the two phrases “I feel good” and “I feel well,” the first is the correct one if you’re speaking of your state of health (physical or mental). ‘Feel‘ here is a ‘linking verb‘ and is followed by a predicate adjective. So if you mean that your health is good, your spirits are high, and your outlook is optimistic, say “I feel good.”
“On the other hand, if you use ‘feel’ in its literal sense of touching something, like feeling for a light switch in the dark, say, ‘I feel well.‘”
(William and Mary Morris, Harper Dictionary of Contemporary Usage. Harper & Row, 1975)
“Today virtually everybody agrees that both good and well after feel and look are predicate adjectives. The years of disagreement over which was correct seem to have contributed to some differentiation. Look well and feel well tend to express good health. Feel good can express good health or it can suggest good spirits in addition to good health. Look good does not generally refer to health, it relates to some aspect of appearance.”
(Merriam-Webster’s Dictionary of English Usage, 1994)