I am not a teacher.
"I feel well" means I am in good health. So does "I am well."
"I feel good" means I am happy/positive/excited, etc...
"I am good." is often heard in AmE to mean "I am well" but in BrE it means "I am a good person".
"I feel bad" is the opposite of "I feel good".
"I feel badly" should only be used when you really want to use the adverb "badly" to modify the transitive verb "feel".
So you don't feel badly for people, you feel bad for them.
Student or Learner