Young's double slit experiment is what can prove light is a wave. He projected light through walls with two narrow slits in them. The result on the wall behind is interference patterns which shows that when the waves of light go through the slits they interfere and cross each other. This also shows diffraction which is the bending of waves around obstacles or spreading of waves when they pass through an opening. Another thing that proves light is a wave is red and blue shifts. When a star is moving very quickly at us it appears to us bluer than it actually is because the wavelength decreases.
The Compton effect proved that light is a particle because it shows that light has momentum.The photo electric effect is when light is shined at a thin piece of metal the photons knock electrons out of the metal. This shows that electrons are pieces of matter because they have momentum.