"Je ne vais pas me sentir bien."
Traduction :I am not going to feel well.
I read "good" instead of "well" but Duolingo accepted this answer. Which answer is better?
I feel good means you're happy about your life, you're satisfied...
I feel well means you are well, healthy
In this context, in French, the sentence has the "health" connotation. So, in this context, the best translation would be I'm not going to feel well. But both translations make sense :)
Normally in English, you use an adjective with the verb "feel". For example, you say "I feel happy" NOT "I feel happily".
The confusing part is that "well" is both an adjective AND an adverb. In this sentence, it is being used as an adjective: it means "in good health; sound in body and mind". The word "good" can ALSO mean "in excellent condition; healthy".
So you can use either "well" or "good" here. They are both correct. Also, as lazouave said, "I feel good" can mean other things (see the dictionary entry for a list). It depends on context! Bon courage !