'... would rather do/be ...' normally expresses a preference for a present or future situation.
I don't like living in the country. i'd rather live in a big city.
I am bored with this assignment. I'd rather be on the beach.
Do I have to go to Aunt Mary's tomorrow? I'd rather stay at home.
'... would rather have done/been' normally express regret about a past situation:
I had to go to Aunt Mary's yesterday. I'd rather have stayed at home.
Student or Learner