I recommend Amazing Grace as an early tune to learn. Whatever tempo you play usually sounds great -- the slower the better. Pauses as you try to remember what comes next just make it sound more dramatic and heart-rending. It also has a very limited range -- a simple version only uses one octave and 6 notes (degabd).

Shenandoah and the Water is Wide also sound great slow and moody (i.e., picking your way slowly through practices).