Studying the replies here again, this time to attempt to understand how C# becomes A, and where we see that on a physical Circle of 5ths (if we do). So, a thanks again for having all this here, that I could come back to it all as needed!
I'm going to write out all twelve chromatic notes for every scale, and try out the other tips posted above, to see if that helps me get it.
I think you have a misunderstanding. Not all scales have twelve notes. For example, major scales have seven and pentatonic scales have five. I do think writing out the notes to scales is useful. Equally important, to me, is being aware of scale degrees (i.e., what's the first note of the scale, the second note, etc.). If you know these things, then you can transpose. For example:
C# major=C#, Eb, F, F#, Ab, Bb, C
A major=A, B, C#, D, E, F#, Ab
If you have something written in C# major and you want to turn it into A major, you just swap notes of the same degree from the two scales. Every time you see an F (3rd degree) in the original song, you swap it for C# (the 3rd degree of the A scale).
I know this sounds complicated, but it gets easier with practice. As a matter of fact, nowadays when I write down music, I never write the actual notes; I write roman numerals indicating the scale degree. I do this so that I can play the music in whatever key I want.
Lastly, I should say that I'm not a musician...so I may have misspoken or made an error. Hopefully others will chime in and correct me if I'm wrong