That’s the neat thing. The speed of light is constant. It doesn’t change. It’s always 1c whether you’re traveling at +1c, - 1c, or 0c. Buckle up for some relativity. The wavelength can compress or expand, but it always travels at 1c.
Let’s say you’re on a ship capable of moving at any speed between 0c and 1c. You’re passing a particular star and want to travel to a planet 1ly away. You have a powerful laser and the other planet has a powerful telescope to detect it. There are calibrated timers on both the planet and on your ship that are synced to each other. .
T minus zero. You flash the laser at the planet as you fly at 0.5c, or 1/2 lightyear per year. The light travels at 1c, or 1ly per year.
1 year after the flash, the planet sees the flash. It traveled 1ly in 1 year. 2 years after the flash, the planet sees your ship arrive. All is normal so far.
From the ship, you know the light traveled at 1c away from you. You arrive at the planet 1 year after the flash, according to your on board timer. One. The light took half as long as you.
Time is not constant, c is constant. The faster you go, the slower time passes. In 1 year of fast travel, you arrive 2 years later, according to the stationary planet. So all of the light physics apply the same, no matter the speed. Time dilates to make up the logical difference. If you reach 1c, time effectively stops and you arrive instantaneously, from your perspective. When we look up at the Andromeda galaxy, some 2.5 million lightyears away, the light we see was emmited 2.5 million years ago - from our perspective. If we see a star go supernova in Andromeda, it happened 2.5 million years ago. But those photons of light, created by a star that died 2.5 million years ago, experience no time passage at all. They instantaneously go from the star to your retina, from their perspective.
That’s basically why lightspeed travel is effectively impossible within our current models. Traveling faster is out of the question because none of it makes sense. It’s not a simple matter of making a new model or believing scientists are idiots. There are many experiments that hold true to the model (such as the atomic clocks used on a plane to test the effect of speed and gravity on time dilation) as well as satellites using the current model to maintain time accuracy. The energy required to get to those speeds is not even remotely feasible. The fastest man made object at 450,000+mph, the Parker solar probe, is still in the 0.0005c range. We tried our best and it’s still just a tiny fraction of 1c. And that’s by using some gravity slingshots and spiraling down into the sun’s gravity well, nothing about leaving the solar system. The Voyager probes that slingshotted out of the sun’s gravity well are down to under 40,000mph.