Suppose a copy of Earth. This planet can travel at any arbitrary speed, and it's set to travel along the galaxy's spiral arms. It rotates counterclockwise, at the rate of 24 hours a rotation. It keeps its standard gravity and atmosphere regardless of its travelling speed.
The observer is standard average human, standing at a fixed point on the equator, using their own naked eyes. Clear sky, no clouds, no light pollution, no loopholes. With rotation, our observer will first pass through the side facing forward, from midnight to midday.
Now I'm going to make some assumptions, feel free to correct them if they're offbase.
If an object moves fast enough, I'm pretty sure that blueshifting and redshifting should happen.
What I expect (or perhaps wishfully think) is that over the course of a rotation our observer would see the stars get gradually bluer until about 6am, then whiter until midday, then redder until 6pm, then whiter until midnight.
What's the minimum travelling speed for such blue/redshifting to be noticeable and visible to our observer?
The question is about the travelling speed (of the planet going forward), not the rotation speed (which is fixed at 24h/rotation). Answers prefered in fractions or percentage of $c$, precision to the order of magnitude is sufficient.
Edit: Here's a diagram from @Tortliena that illustrates the situation