In the mode line, emms shows the full path to the file being played, how do I change that?
emms will only do this if it doesn't have any more information about the track. If your track has meta-data (for example id3 tags for mp3 tracks) then emms can use this if the right parts of emms are loaded. This should happen by default if you have the backend programs installed.
Check the "Track Information" node of the emms manual for more details and to see what backends are needed.