There is no real right way to listen to music, anymore than to engage in any other human activity. This is the problem with courses in Music Appreciation which often give the impression that music is made up of distinct themes and procedures (e.g., fugue and sonata form) and that a respectable listener should be able to track these things.
A high school student once confided to me that in order to pass his Music Appreciation course he had to differentiate two themes in a composition though it was clear to him that one was a variant of the "other."
I have a good friend who has the ability to identify sounds as they flit past, including a vivid awareness of harmonies and overtones. It is a skill I do not possess and do not envy. I much prefer letting my being be carried by music wherever it leads me--sometimes the most familiar music leading to the most unexpected experience.
If a piece of music does not hold my undivided attention either the composer or the performer has not done their job.