□ feel good : 기분이 좋은
A : How are you feeling today?(오늘 기분이 어때?)
B : I am feeling good. (기준이 좋아!)
□ feel well : 건강이 좋은
A : How was your cold? (감기 어때?)
B : I am feeling well. (좋아)
"Feel well," usually refers to a good health condition.
"Feel good," on the other hand, can also be used in this way, but is also very commonly used to refer to a good emotional condition.
("feel well"은 보통 건강이 좋다는 의미이고, "feel good"은 보통 기분이 좋다는 의미이다.)
(Someone sick long in the hospital says to the person visiting him/her.)
"I can't feel well but I feel good this morning."
-- You seem pale and weak. Are you ill?
-- It's nothing really. I feel fine[well].
"I feel good" and "I feel bad" are both perfectly good English sentences.
These normally have to do with your mood.
There are also:
"I feel well" and "I feel ill".
These have to do with your health.
"I felt ill yesterday, so I went home from work early.
Today I feel well." (or "Today I feel better.")
And also:
"I feel that I am a good person."
"I feel that ..." is different. It is a way of saying "I believe that ...".
As such it does not have anything to do directly with feelings of mood or health.
"I feel good" cannot mean the same thing as "I feel that I am a good person".