According to grammar rules the modal auxiliary shall is used in the first person and will in the second and third persons to talk about future.

I shall start the work tomorrow.

The future auxiliaries will and shall are often replaced by other verbs and expressions that stress a particular feeling or attitude of the speaker.

Modal auxiliary verbs can express various aspects of obligation and freedom. Note that the use of modals verbs can help in making a request, suggestion or instruction sound more polite.

Would is the past tense form of will. It is a modal auxiliary verb. After would, infinitives are used without to.

I would like some advice. (NOT I would to like some advice.)

In indirect speech

Should is the past tense of shall in indirect speech.

The distinction between shall and will is becoming less significant and is now strictly observed only by precise speakers. Shall is becoming much less common with second and third person pronouns. In the first person, however, shall is still being used to talk about simple future.

With the first person

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