We don’t use shall very frequently in modern English, particularly in American English.
“Shall I” or “shall we” is used in questions to make offers and suggestions, also to ask for advice. As in “Shall I open the window?” – it simply means “Do you want me to open the window?” (You give a suggestion). Or in “I’ve got no money. What shall I do?” – it simply means “I’ve got no money. What do you suggest?” (You ask for an advice).
• What time shall we meet?
• Shall we vote on it now?
• What dress shall I wear?
• Let’s go, shall we?
Pay attention to the difference between “shall I” and “will you” used in the following sentences:
• Shall I close the door? (Do you want me to close it?)
• Will you close the door? (I want you to close it.)
Formerly, in older grammar, shall was used as an alternative to will with ‘I’ and ‘we’. Today, will is normally used. When we do use shall, it has an idea of a more personal, subjective future.
• I shall go to see the boss and I shall ask him to explain this decision.
Notice that the negative of shall can be shall not or shan’t – though the second one is now very rare in American English.
• I don’t like these people and I shall not go to their party.
• I shan’t object if you go without me.
• You shall not pass the exam unless you study hard.