There are two different issues. One is that light travels at the same speed no matter how you move relative to it; this surprising result is accomodated by the nature of the space and time it moves through. The other is the straightforward issue that the time difference between successive waves does what you’d expect due to relative motion, which causes the Doppler effect.
Radio or light signals passed between rocket ships as they change their relative velocity will always hit the receiving ship at the same velocity relative to the ship (imagine a photon speedometer on the ship - it will always catch the signal and register the same speed, never faster or slower). It will also be at the same velocity relative to anything else (which is very much the point of special relativity).
The time frequency or wavelength of the received signal will depend on the different speeds involved. Intriguingly it is not necessarily what you’d calculate based on the present speed of the two rocket ships, nor what you’d calculate based on the speed of the emitting ship at the time of emission and the speed of the receiving ship at the time of reception. It is complicated.
Viewing an object which is fleeing you, you would see its appearance modified by redshifting. That is, the wavelengths are all increased, relative to what you’d see if your relative velocity were zero.
It is somewhat harder to picture because we have a mental concept of “now” that extends everywhere. We might think, if someone here on earth and someone else on the moon set off a camera flash at the same instant, etc etc. But things that happen at different locations don’t have just one version of relative timing. After you account for how long it takes to get a signal from here to there, you will still have a difference between the two different “nows” that depends on how you are moving. It isn’t an artifact of how you are doing the observing - the time and space components are intermingled in a way that depends on velocity.