I'm a U.S. native speaker and these are comments about U.S. usage. This is the sort of detail which might be different in the U.K. "I need to change my appointment" is fine, clear, natural-sounding and a native speaker might say it exactly that way. You don't need to read further unless you're interested in other ways to say it. "I need to reschedule my appointment" is slightly more formal. "I need to postpone my appointment" gives slightly more information, because it says you are asking for a later time than the original appointment. "I need to change/reschedule/postpone the time of our next lesson" is another way to say it. One has a "reservation" for a hotel room, a rental car, or a plane flight, so one would say "I need to change my reservation."

I say 'reschedule', e.g. I need to reschedule my appointment/my lesson/my meeting with you