We don't use 'Shall' very frequently in modern English, particularly in American English.
It is used to make offers and suggestions and to ask for advice.
What time shall we meet?
Shall we vote on it now?
What dress shall I wear?
Shall I open the window?
You only really need to know that about 'shall' in modern English. Read the rest of this only if you want to know more about how some older speakers still use 'shall'.
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.