Doppler effect and constant velocity of a wave

Light sometimes behaves like a wave, sometimes like a particle, but for most people the idea of a photon being a particle is misleading because we tend to think of a particle as having mass and taking up space. Because a photon has no mass it is required to travel at light-speed (which is actually the speed of causality). It does not gain velocity - from the very instant it is emitted it is going at the fastest possible speed.

Your question and objections make complete sense in a Galilean world, a world of 3 orthonormal space dimensions where time is independent and universal: that is, all observers agree on the same clock (and ruler). In such a place, all velocities are additive. (Note: our slow moving daily lives are approximately Galilean--and Galilean transformations can be used to transform reference frames).

In reality, we live (locally) in Minkowski space, where the space dimensions mix with the time dimension, and different moving observers have different clocks and rulers. One needs Lorentz transformations to connect reference frames.

If find the number one stumbling block to understanding the latter is the lack of simultaneity. Moving observers cannot always agree on time ordering, and forgetting this leads to many paradoxes (apparent, of course).

In particle physics, photon is one of the fundamental ingredients of the Standard Model (SM) which explains all the particles/fields and their interactions. A symmetry group particularly $SU(3)_Ctimes SU(2)_Ltimes U(1)_Y$ governs all the properties and interactions of those particles. In very small distances ($lt 10^{-18}$m) or at very high energies those particles are all required to be massless by the theory, at low energies however some of those particles acquire masses by a mechanism called symmetry breaking and photon associated with electromagnetic interactions stays massless. In Relativity, massless particles are destined to travel at the speed of light which is the ultimate limit for matter particles. That's why, for example, the range of the electromagnetic interactions is infinite. The answer of your question thus lies in 4-dimensional relativistic structure. As JEB already mentioned above that Lorentz transformations of 4D spacetime is quite different than Galilean transformations of 3D space.