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.