I know the colours of the stars relate to their heat, however if I were standing on a different planet identical to Earth, but orbiting a blue Bright Giant or a Red Giant, or perhaps even a Red Dwarf, what colours in visible light would I be able to see? Is it the same as with our star, i.e. would any be missing such as if I'm in a room with a black light?
Compare the effective temperature of the stars with Planck's law, and with a light bulb (2000 to 3300 K): The color of a light bulb corresponds to spectral class L (red brown star) or class M (red star, including dwarfs and giants); violet, and some blue are missing.
Light of Brown Dwarfs (spectral class T) would look red from their black body radiation, but due to their chemical composition the actual color may vary a bit over different tones of red. They would mainly feel hot, and look dull red at the same time, similar to glowing iron. Colors can hardly be distinguished under these lighting conditions.
Some of the very hottest stars (spectral class O) may ressemble "black" light, looking bluish, and causing fluorescence due to to the shift of the spectrum into the ultraviolet, including x-rays. Without protective ozone/atmosphere layer, this wouldn't be healthy.
This may apply to some degree also to spectral class B (blue white stars); plasma cores of lightnings are of a similar temperature (up to 30,000 K or more) as the effectve temperature of class B or class O stars.
Any spectral class in-between (A, F, G, K) would look almost white, a little bluish for hot stars, yellowish for cooler stars, after some adaption of the eyes.