The colors don't change, but you'll see a red or blue shift in the spectrum based on if the planet/star/galaxy/object is moving towards you or away from you. If it's moving towards you, there's a blue shift, meaning the wavelength decreases due to it stacking on top of each other (Doppler effect), redshifts mean that it's moving away, and the wavelength is stretching. It's not really something we see with our naked eye because the shift is quite subtle. We use Doppler spectroscopy to help detect planets around stars, but in terms of finding their distance, we go off the host stars.
Typically we use a stars stellar parallax to find the distance to a star. How we do this is first measure the stars position, and then 6 months later (when we're on the opposite side of the sun) we measure it again. The star will have moved slightly from our position. We take the angle of that movement, which is measured in arcseconds and determine the distance with the formula of d=1/p (distance equals one divided by parallax). That gives us the distance in parsecs.
For example, Betelgeuse has a parallax of about .005, so 1/.005 = 200. Betelgeuse is roughly 200 parsecs away, or 652 lightyears. However in reality, when we make the number smaller, we get 197 parsecs (643 light years). The smaller you can get the parallax, the more accurate you can get that distance, however it's very difficult to do that. Even now, Betelgeuse is considered 643 light years, ± 146 light years, which is still pretty big range.
The other way we determine the distance is by measuring the brightness of a star (by the color of the star), and then measure the actual brightness to the apparent brightness we see from earth, but this is less accurate.
If you were to travel at the speed of light, time would slow for those traveling, but would continue moving at normal speed for others. Traveling at light speed wouldn't let us arrive in the present (meaning how the planet actually is right now), but those who traveled at the speed, it would seem quicker than those who aren't going that fast.
Essentially if we were to travel at 99% of light speed, it would take just over 1400 years to get to the new planet. However for those on the ship, it would feel much quicker. Well rather, it would feel normal, but you would be moving quickly. If my math is correct those on the ship, it would feel like they've traveled for about 6 1/2 years on the ship (that's how much they would age), but Earth and Kepler 452B would age 1400 years. It would also take 1400 years (2800 years round trip) to relay messages back and forth. So essentially it would take NASA 2800 years to figure out if they survived the trip, and the travelers 2806.5 years to figure out if NASA still exists.
