Saunter, stroll, amble can all mean to walk slowly and more or less aimlessly, especially in the open air.
Saunter suggests a leisurely pace and an idle and carefree mind.
Stroll differs from saunter chiefly in its implications of an objective (as sight-seeing or exercise) pursued without haste and sometimes with wandering from one place to another.
Amble occasionally conveys the same implications as saunter or sometimes stroll, but it more often suggests merely an easy, effortless gait comparable to that of an ambling horse.