'Shall' is fairly rare in modern English, particularly in American English.
We can use it to make offers and suggestions and to ask for advice.
Shall I open the window?
Shall we start the meeting?
What time shall we start?
What dress shall I wear?
That is really all you need to know about 'shall' in modern English. Only read the rest of this if you want to know more about how some older speakers still use 'shall'.
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 will find different grammar exercises and more grammar explanations on our sister site English Grammar Secrets