Shall is usually followed by an infinitive without 'to': I shall explain everything later. Sometimes it is used without a following infinitive: I have never visited America and probably never shall.
Shall does not change its form, so the third person singular form does not end in '-s': The President shall appoint all ambassadors with the consent of the Senate.
Questions and negatives are formed without 'do': Shall I come with you?
The negative form shall not can be shortened in conversation or informal writing to shan't.
Shall has no participles and no infinitive form. Shall can be used for forming the future tense of another verb when the subject is 'I' or 'we', but it does not have a future tense of its own: We shall see you tomorrow.
Should can sometimes be used as the past tense of shall, for example, in indirect speech introduced by a verb in the past tense: I hoped that I should not need to defend myself.
Shall have followed by a past participle forms the future perfect tense and shows that an action will be completed before a particular time in the future: We shall have completed the first section by Monday.