It occurs to me when I want to say I am gonna go out to some place. Can I say I'm gonna go away to some place? Do these two phrasal verbs make the same meaning?
They're not interchangeable.
To go out implies somewhere relatively close, that you're going to return from sooner rather than later.
You can go out to work, out to a bar, restaurant or club.
You would more likely go away to a further destination.
You could go away to Paris, London or Rome - even if just for the day, you wouldn't say 'go out to Rome'... though once you went to Rome, you could go out for the night.
You could go away to work - but that in itself would imply you were going to Paris, London or Rome in order to do that.
If 'going out' implies a brief excursion rather than a distant stay, then perhaps it's partly psychological rather than purely geographical.
< sarcasm >
If in the latter days of the British Empire/Raj you could consider 'going out' to India - of course you'd be back after several G&Ts & have some tales to tell of how it was always so damn hot, Caruthers - then these days you could equally consider 'going out' to Portugal for the 'season', or to wherever they have the good stuff for skiing in the winter, with après ski by the pint ... like some modern-day debutante.
Papa, of course, will pick up the bill...