God, grant me the serenity to accept the things I cannot change,....is that it doesn't offer any help with "knowing the difference". And that's the most important part. So here's a way to recast the whole thing:
Courage to change the things I can,
And wisdom to know the difference.
Every problem is either 1. a problem in the world, or 2. a problem with your perspective.
Hint: it's nearly always the latter. Problems of perspective (e.g. this) account for 99% of perceived problems in the rich world. We mentally manufacture problems to fill the vacuum of actual ones (we do this to ballast our happiness).
How to distinguish? Easy. If the problem is truly in the world (e.g. bullets suddenly start firing, or your stomach growls from emptiness, or a child starts wailing), you don't need to ask yourself whether there's a real problem. Real problems spur action, period. Problems of perspective, however, make you think about it.
It's a lot like this diet tip: "No one in the history of the human race has ever asked themselves 'Do I want to eat some more?' while still hungry. Genuinely hungry people just eat! The fact that you're asking means that you have, in fact, eaten enough!"