Would is the past tense of will in indirect speech.

• He said, ‘I will come.’
• He said that he would come.

Should is the past tense of shall in indirect speech.

With second and third person pronouns, would is used to express a wish or preference.

• He would rather stay at home than work.
• She would rather have tea than coffee.

Should can be used with first person pronouns to express a wish or preference.

• I should like to go to the pictures tonight.

Would can be used with 2nd person pronouns to make a polite request.

• Would you mind opening the window?
• Would you mind moving a bit?

Should can be used with nouns and pronouns of all persons to express an obligation or an assumption.

• You should be more punctual.
• They should be more considerate.
• It should be possible to do it.
• They should reach home any time now.
• Should have + past participle can be used to talk about past events which did not happen.
  • I should have phoned John this morning, but I forgot.
  Would can be used to talk about past habits and typical characteristics.
  • Sometimes he would bring us little presents without saying why.
  Stressed would can be used to criticize people’s behaviour.
  • She was a good girl, but she WOULD talk about herself all the time.