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 anytime 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 behavior.
- She was a good girl, but she WOULD talk about herself all the time.