It's a social and cultural question. It depends on context. It depends on the level of formality. It depends on how serious the error it is and how easy it is to correct. It depends on whether you are trying to be kind, or whether you are annoyed or angry.A
"Mistake" and "error" are good, neutral, factual words that can be used on all situations. You can use them, and let your tone of voice and body language take care of emotion.
"Goof" is an informal word, and it is fairly soft. A "goof" is a minor mistake, of the kind we all make frequently. "There's a goof in this caption," or "You made a goof here."
"Messed up" is stronger. It's a criticism. It's something that shouldn't have happened. And if something is "messed up," it need to be "cleaned up," i.e. corrected.
"Screwed up" is stronger, and suggests that the speaker is angry.
I don't think a foreign speaker should try to use taboo words, even if you hear them all the time in songs and on TV shows and YouTube. It takes a good understanding of a culture to know when it is safe to use them.