I got into Cash about 6 months before the films release, but more specifically the American Recordings sessions he did with Rick Rubin. The cover of NIN's "Hurt" was the initial attraction but within these releases there are a bunch of re-hashed old cash songs, new cash songs (which are stellar) and some of the best covers you will ever hear, most notably Depeche Modes "Personal Jesus", Will Oldham's (Palace) "I See A Darkness" and Nick Cave's "Mercy Seat".
These are highly recommended releases but its also wise to check out his earlier stuff just because its classic essential listening for anyone with an interest in good music.