Nach vs Nachdem?
According to Duo both mean 'after' but nach has the additional meaning of 'to' in regards to going to countries and cities (maybe states too). But just in the meaning of 'after' how do these do differ?
"Nachdem" is for when you want to say "after" with a whole phrase, like "Nachdem ich nach Hause gegangen bin, habe ich geschlafen" - "After I went home, I slept." "Nach" is for saying "after" with a noun, "Nach dem Frühstück gehe ich in die Schule" - "After breakfast I go to school."
In terms of grammar, it's like the difference between "during" and "while" in English. You use "during" with a noun/pronoun, and "while" if there is a whole clause with a subject and a verb:
He fell asleep during the movie.
He fell asleep while we were watching the movie.
Similarly, "nach" is used with a noun/pronoun ("nach dem Film" = after the movie) and "nachdem" is used if there is a whole clause with a subject and a verb ("nachdem wir den Film gesehen hatten" = after we had watched the movie).
TEST: You have to use "nach" if you can grammatically replace it with "during". You have to use "nachdem" if you can grammatically replace it with "while". (The meaning changes, of course, but this is just about grammar. And it only refers to cases in which nach/nachdem are used in the sense of "after").