I usually find it useful to give concrete examples of the philosophy that cosets are what's left when one wants to neglect the elements of the considered subgroup.
The most basic example is the following: turn the light of the room on and off repeatedly and fast a number of time, and then ask the students: "how many time did I hit the button?" You want to have done it in a way that they won't be able to answer. Then ask: "did I hit the button an even number of time or an odd number of time?", and then the answer only depends on the state of the light before and after the operation.
The point is that the light does not care exactly how many time you hit the button, it only cares if this number is even or odd, or in other words the light does exactly as if 2 where equal to 0. The state of the light is described by a quotient, whose elements are cosets (all the numbers of it inducing a given state of the light).
Lights you tap to cycle through dim light/bright light/off give a similar example.
One possibility for orbits with a non-abelian group is to find several standard and a non-standard dices (most dices have the numbers arranged in the same patterns, but not all of them). Ask your student if two dices are the same : they will try to rotate them so that they match. The point is that not any position will do, and if they succeded it means they applied an element of the group of displacements of the cube (aka $S_4$, but that is another story of course) to a labeling of the faces by $1,\dots, 6$, to obtain another labeling. If they (provably) can't, then it means that the two dices belong to different orbits. The point again is that how to try answer the question should be obvious to the students. Then one can frame their methods in the mathematical framework and hope that they make the connection between what they did and the object of the course, and then the later may start making sense.
The problem with this example is that the set of labelings has no natural group structure, so it does not fit your question well. I guess there are other possible examples which may do.