The axes I'd animate on the wing controls of a hypothetical rig.
Describing these bone rotations can get a little confusing, I think it might help if I show a visual representation of them. By creating a hypothetical rig of a bird's wing we can see which axes we should animate to accurately mimic reality.
It might be preferable in production to create a IK wing rig but for simplicity I've shown a straight forward FK wing where all the bones in the wing would be represented an by an animation control.
The Humerus can rotate in all axes. You could put all rotations on the same control but I find you can avoid rotation gimbal by creating an offset control here and keeping some of the rotations separate. i.e.z rotations on a primary control and y and x rotations on an offset control.
I try to animate the Ulna and Radius control in just one axis, rotation y, as that is the only axis these bones can rotate in life.
The carpometacarpus' z rotation is what causes the main bend in the wing as the first joint rotates up in flight...
.... but by adding a little rotation z to this second control, you can create a curve at this point in the wing and stop the bend appearing as harsh. The first wrist joint above can rotate in y but this second joint cannot.
I animate the feathers along their length for drag, I usually find that just this rotation on the feathers is enough for an initial cycle, then I'll often add further rotations in the shot to break them up. Also, a sense of the feathers spreading on the down-stroke can be achieved by animating their base along the y axis.