How to convert Apple Music M4P to MP3?
M4P files are protected with Apple's Digital Rights Management (DRM) to stop conversion etc. Any unprotected MPEG4 files like .m4a can be converted easily with iTunes or various other applications but the DRM prevents this. 

If you've purchased the music yourself, then upgrading to iTunes Match will let you replace the DRM protected files with standard m4a which Apple now uses for iTunes purchases. It's an annual subscription but you might also find it useful for syncing and updating other music in your library.

Also bear in mind that any conversion from m4a to mp3 will be from one lossy format to another, so you might notice a reduction in sound quality.