In waves particles don't have to travel. For sound waves, they oscillate back and forth with pressure changes in the air, or solid/liquid materials. For water waves, water molecules move in orbits, you can tell this because when water waves hit the shore, the water level doesn't instantly change, they don't fill the shore, so the molecules move back.
When you say material waves are 'made of' something, it's more accurate to say material waves travel through space through a material medium. The particles are transferring energy in the form of a wave.
Light waves aren't 'made of' anything, they are a pair of disturbances in electric and magnetic fields, hence their name, electro-magnetic waves. When an electric field changes, it creates a magnetic field, and when a magnetic field changes, it creates an electric field. To understand how this works, you can look into Maxwell's equations of electromagnetism, specifically, the last two.
The wave-particle duality was discovered first for EM waves, with an experimental phenomenon called "The Photoelectric Effect". When light waves at a certain frequency were shined upon a metallic surface, it was observed that electrons popped out of the metal, and unlike water waves carving away rocks at a shore, this happened instantly, indicating that light had a particle property. Yet at this time, the wave properties of light were already well established, eventually leading to the only possible conclusion, light simply had to be both, that it was made of massless particles called photons that carry energy, had momentum, and also had wave properties.
Later though, this was discovered to be true for more than just photons, all particles have wave-particle dualities, this was -I think- first proven for electrons, when they were made to create an interference pattern exactly like waves do. These particle waves are called de Broglie waves, this is one of the most fundamental ideas of quantum mechanics, therefore it is very difficult, perhaps impossible to really understand intuitively. It's a very shocking way the universe works.
So to fully understand the behavior of quantum particles and light waves, both theories are used, in certain situations. Light wave events like diffraction, interference, polarization, etc. are fully describable mathematically by wave theories, other events like the photoelectric effect, Compton scattering, etc. are explained by the particle theories.