Light behaves as both a particle or a wave, but never both at the same time. It is circumstantial on what the light is doing/what kind of situation it's in. You're correct in saying that light has momentum, however not in the conventional sense. Momentum is calculated by taking the product of the mass and the velocity of an object, p = mv. However light has no mass, hence looking at it from that perspective would not suit light. This class of physics, known as newtonian physics, doesn't work with thinks on a small level, or quantum level. Hence people describe things like light and tiny (quantum) particle using quantum theory and quantum mechanics. Newtonian physics break down (don't work) on a quantum level, and the equation p = mv is an example of newtonian physics and as mentioned above, doesn't work with light, because light belongs to the quantum world, so we have to look at quantum theory to describe it.
In 1924, a physicist called Lois DeBroglie proposed that all matter has a probability wave. In simple english, all matter can behave as a wave, and all waves can behave as matter. The equation he gave for this was:
Wavelength = h/p
Where 'h' is planck's constant and p is the momentum of the object. If you re-arrange this equation for p, momentum, you get: p = h/wavelength. This is an example of an equation used in quantum mechanics and therefore you can see how light can have momentum in the quantum world despite not having mass. As long as its wavelength is known, you can know its momentum, and h (6.63 x 10^-34 if I remember correctly) is a constant so you dont need to worry about that. So you can see from DeBroglie's hypothesis and equation that it is not the speed of light which effects momentum, after all how can it as it is a constant value at all frequencies/wavelength, hence implying that all light at all frequencies has the same amount of momentum. This isn't true, and the momentum is inversely proportional to the wavelength, which makes sense. As wavelength increases, momentum decreases, because longer wavelength is a lower frequency. Energy is proportional to frequency, given by the planck equation, E = hf, again h is planck's constant, f is frequency of light and E is the energy of that light. The more energy you have, the more momentum you have, hence DeBroglie's equation makes sense logically as well as mathematically.
There is so much to wave/particle duality of light, this is just a taste of what scientists are pondering. If you want proof of matter being able to behave as waves (DeBroglie's hypothesis) read about the electron diffraction experiment, it basically showed that one electron, just one, diffracted through a narrow slit to give interference patterns, which made no sense because this implies that one electron had to be in multiple places at once. Explanation? The electron (matter) was interfering, just like a wave would. Experiments proving that waves behave as particles include the photoelectric effect. A lot of reading, but hope you understand most of this, you sound genuinely interested. And forgot to mention, a particle of light is known as a photon, so whenever you see that word, just know that it's talking about light.