What was the first music recording that people could go to the shop and buy a copy of? Was it even a gramophone record, or was it some kind of wax cylinder recording?
It would have been an Edison or Columbia wax cylinder, ca. 1895 or so. The first few years of the phonograph were mostly about using it as a dictation machine, so it was sold with blank wax cylinders. (The first surviving cylinder recording of an identifiable piece of music is from about 1888, but it was an amateur "home" recording of a rehearsal at London's Crystal Palace of a Handel oratorio.) Disc recordings don't start until about 1915