Is there a word for "part of a route"?

If the person using your program is deciding where each of the main stops are, each segment is called a "leg". Most routing software uses this terminology when planning multi-stop itineraries, assuming that the user is driving or otherwise in control of the vehicle. When applied to commercial flights, we generally use the term "leg" for each flight, as well, even though the customer is not necessarily the one making the choice about each stop.

However, we don't generally use the term "leg" to refer to every change in path, every navigation action. Better terms would be a "turn", "step", or, possibly, "waypoint." ("Waypoint" has some of the connotations of choice that "leg" has, however). Most users won't think of the "edge" as the important thing, but if they did, you could call that a "route segment." For most users of turn-by-turn navigation systems, the vertex (turn) is the thing that they focus on, rather than the edge (segment).