With the first person

Shall is used with first person pronouns to express simple futurity.

I shall leave for Texas tomorrow.
We shall discuss the matter with him.
I shall be thirty next Friday.
We shall invite them to the party.
I shall begin the work tomorrow.

In the example sentences given above shall is used to talk about the strong possibility or certainty of an event or action which is going to take place in the future.

With the second and third persons

With the second and third person pronouns, shall may be used to express a command.

You shall not lie. (You are commanded not to lie.)
You shall leave this place at once. (You are commanded to leave this place at once.)
He shall obey my instructions. (He is commanded to obey my instructions.)
You shall not tell this to anybody. (You are commanded not to tell this to anybody.)

Sometimes shall is used to talk about a promise.

You shall be given a present if you study well.
He shall be promoted.
She shall get a prize if she does not make a single mistake.

Shall is also used to express a threat.

You shall regret this.
He shall be dismissed if he doesn’t obey my instructions.
They shall pay for this.

Sometimes shall expresses determination on the part of the speaker.

You shall apologize. (I am determined that you must apologize.)

Will

With the first person

With the first person pronouns will expresses ideas such as determination, promise, threat, willingness etc.

Determination

I will go, whatever happens.
We will not wait.
I will not let him go.
I will study well and pass the examination.

Promise

I will do whatever I can to help you.
I will give you a call after I return.
I will try to get you a job.

Threat

I will teach him a lesson.
I will kill you if you do not obey my orders.
I will punish anyone who creates trouble.

Willingness

All right, I will come with you.
‘There is the door bell’. ‘I will go.’
Of course, I will lend you my car.

With the second and third persons

When used with the second and third persons, will expresses simple futurity.

The train will leave at 10 am.
The school will re-open on Monday.
He will arrive at 8 o’clock.