"Stuff" is a very general and informal word. You wouldn't use it in formal instructions. "Bring your stuff" is something that might be said in casual conversation.
"Kit" means a specific, designated set of items. If you said, "Bring your kit", the listener would understand you to mean a set of items that he had been given for this purpose, or that he was told to gather together. We'd expect everyone to have essentially the same kit. Or at least, that there would be groups with each having their own kit, like if you were talking about a group of medical personnel, there might be a "nurse's kit" and a "surgeon's kit" and an "anesthesiologist's kit", etc.
"Equipment" is more general. Each person might have their own equipment. Equipment is generally understood to mean tools or mechanical or electronic devices. For example, for a team of archaeologists, picks and shovels might be "equipment". Their clothes and personal items would not be.