"shall" has the meaning of "will" but is less common and can sound more formal [usually in first person]: "We shall/will meet again." It can also be used when commanding [usually in second or third person]: "You shall not pass!" And also used in questions, suggesting something to happen now or in the future: "Shall we leave now?"

I would just add that "shall" is not common, in my opinion — except in the structure "Shall we ____?" or "Shall I ___?" Both are very polite. "Shall we go together?" (Very politely suggesting that we go together.) Even more common is simply "Shall we?" (Politely suggesting that we do what we're about to do. I often say this at the end of dinner after we've paid the bill, to suggest that we all stand up. You can also say it when everyone is ready to go somewhere, to suggest that everyone start moving.)

shall = will, used with "I" and "we"

"Should" has a more imperative tone, implying some sort of restriction is in place; "shall" gives the listener more freedom to choose. You may interpret "should", "shall" and "must" (bonus!) as: "Should we dance?" -> "I'm not sure if we are allowed to dance or not, shall we do it?" "Shall we dance?" -> "May I have the pleasure of dancing with you?" "Must we dance?" -> "I don't want to dance, are you going to force me to do it?" "I shall go now" -> "I will go now, it is my choice" "I should go now" -> "I think I will go now, or else something bad will happen" "I must go now" -> "If I don't go now, aliens will definitely come and abduct me" The phrase "you shall not pass"/"thou shalt not pass" would be confusing to you, as it implies "you may not pass (as I am currently preventing you from doing it)" more than "you are not allowed to pass (because of the rules)".