I haven’t modded on Lemmy so not sure how applicable these all are.
When removing a comment, cite the specific rule they are in violation of. It helps them understand why, it helps anyone else in the community know why, and it helps yourself and other mods if the person appeals down the road. Be transparent.
When removing a comment because a portion is unacceptable, i.e., they throw in an unnecessary slur when the rest of the comment is fine, offer to restore the comment once they edit it to remove the slur.
Establish ban guidelines within the mod group. First time offense? One day. Egregious offense? What’s the threshold for “egregious” and when does it result in permaban? This person who got a 1-day ban for the f-slur, is the same ban length being consistently applied to other users of the f-slur?
If you feel a personal dislike of a user, check your decisions against another mod for an outside perspective. Or better yet, just ask them to mod that particular comment/post and remove yourself from the conflict of interest.