Form and rules
How it works
Would rather + base verb: I'd rather stay home.
Would prefer + to-infinitive: I'd prefer to stay home.
Use would rather A than B; with prefer, use prefer A to B or would prefer to A rather than B.
For another person's present/future action, use would rather + subject + past simple: I'd rather you came later.
Negative preference: would rather not + base verb / would prefer not to + infinitive.