Experimentally one never observes waves. Light is detected based on its interaction with electrons and that is always by an electron absorbing a quanta of energy, not via some continuous process as would be the case with waves.

Classically one can imagine that as if electron was hit by a particle. But then we have light diffraction and interference, which classically is described as a wave. So from a classical point of view light travels as a wave but interact as a particle.

As of nature of the light, then consider that there is a reformulation of a classical electrodynamics that eliminates electromagnetic waves all together. There are only electrons that interacts with each other directly with no waves in between. Feynman spent quite some time trying to develop quantum electrodynamic based on that. He failed. Still the point stands that we never observe light directly but only through its effects on electrons and other charged particles. So it could be that what we call light is a theoretical artifact and there is no light in reality.