Hi Andrew,

Great question! The C major chord is generally taught first because it's laid out in a way that's easy to understand, like the white keys on a piano. But you can arrange the chord notes in any order, in fact this is called a chord inversion.

Generally "Power Chords" are arranged in 1,5,1 order and barre chords are arranged in 1,5,1,3,5 order, like the A chord you described. Or, if it's an E chord shape (root on 6th string): 1,5,1,3,5,1

It's really fun when you start mixing the order of chord notes! :D