Would it be correct to say "Je suis retourné me coucher après que tu partis"? I'm confused re when to use passé composé versus passé simple.

Don't fret about passé simple. It is never used unless you're writing literary French. Did you mean another tense? Maybe plus-que-parfait?

You use the plus-que-parfait if you want to stress that the two actions are temporally farther apart. Either one is correct, but the choice depends on the temporal relationship between the two actions.

Je suis retourné me coucher après que tu es parti. -- I went back to sleep after you left.
Je suis retourné me coucher après que tu étais parti. -- I went back to sleep after you had left. (This doesn't quite work for me in English, though).

P.S.: In my first post there slipped an extraneous "s" into "partis" that shouldn't be there.

Je suis retourné me coucher après que tu es partis. --
I went back to sleep after you left.

You need the passé composé.

Thanks Chris.  I understand the rule, but I don't have a clear understanding of when to use passé composé and when to use passé simple.

Thanks Chris, that's really helpful!

You're welcome!