Well it’s kind of complicated. First, for the stars you can see with the naked eye. The ones that look red are cooler, and the ones that look blue are hotter. Indeed you can figure out their temperature approximately just by noticing their colour.

The main colours of stars which you notice with the naked eye are red, white or blue.

• If the star looks blue - a bluish white, its surface temperature is around 7,500 K (7,200 C, 13,000 F) or hotter
• If it is white its surface temperature is about 6,000 K (5700 C , 10,000 F)
• If it is red - a pinkish white really - its surface temperature is about 4,500 K (4,200 C, 7600 F) or cooler

It’s for the same reason that as you heat metal it goes red hot, then orange, then yellow, then white hot, then finally a bluish white in colour. Same happens to stars. BTW as you heat up a piece of metal, it never goes green. You’d think it would go through all the colours of the rainbow, but it misses out green. Same is true for stars. There are red, orange, yellow, white and blue stars. But no green stars. For the reasons, see Why are there no green stars? - Bad Astronomy. In fact our own star is as close to green as a star can get. But it looks white.

Note our Sun doesn’t actually look yellow, that’s just an artistic convention, for instance in Japan children paint the sun red for a similar reason, the Japanese convention is that the sun is red. For explanation of that, see my comment below

So, a red star will probably be a very cool star. But it’s just the outside that has to be cool. A star may be much hotter inside than outside.

There are two main kinds of stars that are cool outside. First, a star like our sun, when it runs out of hydrogen (which our sun is burning to helium), starts helium burning. Helium burns much hotter than hydrogen - which rather paradoxically, causes the star to expand and its surface to cool down. It’s got helium burning hotly in its core, but it’s got hydrogen outside still - and the helium burns so much hotter than hydrogen that the whole thing gets much much larger. So large that when that happens to our sun, the surface of the sun will stretch as far as Earth’s orbit.

This shows how much larger our sun will become as a red giant, compared to what it looks like now.

Note, though our sun is often shown as yellow in diagrams, and eclipse filters and such like often tinge it yellow or orange, and though astronomers even call it a yellow dwarf, our sun is actually a white “yellow dwarf” rather than yellow.

Our sun is that tiny dot at bottom left of the picture. As a red giant it will be so big, that our Earth would skim through the Sun’s very tenuous outer atmosphere if it stayed in its present orbit, and wouldn’t last long. However our sun will also shed a lot of mass in the beautiful “planetary nebulae” as it becomes red giant. So Earth will probably orbit a fair bit further out and may escape getting incinerated.

All this is billions of years into the future. Our sun is still very young for this kind of star and will stay in its stable middle aged phase for hundreds of millions of years, hardly changing at all, and it will be billions of years before it goes red giant.

Earth will be far too hot for life by then, unless of course, there’s some civilization that puts up lots of sun shades in space to cool it - or perhaps - moves it further out. This is billions of years into the future. So probably many intelligent species, not just civilizations, have come and gone since then - or if we are so lucky as to be still around by then, continuing as a technological civilization, our civilization surely is very advanced and could do almost anything.

Anyway - so some of the brightest stars in our sky like Betelgeuse are red giants. They are actually a long way away, because they are so bright. Though their surface temperature is low, not much hotter than an oven, they are so very large that they are also very bright as seen from a distance. So we can see them a long way away. Betelgeuse, top left shoulder of Orion, 642.5 light years away, is one example. Aldebaran is another, an orange giant, 65.3 light years away.

However other stars start off cool, and stay cool, just because they are so small. They burn hydrogen like our sun, but they are nowhere near as big as our sun. Those are the Red dwarf - Wikipedia stars. The closest red dwarf star is Proxima Centauri

“Relative sizes of young stars from the smallest “red dwarfs”, weighing in at about 0.1 solar masses, through low mass “yellow dwarfs” such as the Sun, to massive “blue dwarf” stars weighing eight times more than the Sun, as well as the 300 solar mass star named R136a1″

Again our Sun is shown as yellow, as so often in these diagrams. It’s actually white.

So the red dwarf stars are very red, but as you can imagine, being far smaller even than our sun (which is a so called “yellow dwarf”) and also being cooler, they are not easy to see. Indeed none are visible to naked eye. The nearest ones are still so faint that you need telescopes to see them at all. Fifty of the sixty nearest stars to us are red dwarfs.

So, even though they are by far the most numerous star in our galaxy, or indeed in the universe as far as we know, they are so faint that not a single red dwarf is visible to the naked eye from Earth.

Amongst the brightest stars are the blue-white super giants. They are young stars that are also huge. The bluish white Deneb is one of them, and though it’s quite a bright star, it’s one of the more luminous of the visible stars. It’s shining at us from a distance of around 2,600 light years. List of most luminous stars. So some of the bluest stars you can see with the naked eye are also very distant, thousands of light years away. You can’t really tell much about how far away a star is from its colour unless you know a bit more about it, not for the ones visible to naked eye.

You are probably thinking about the red shift. That though only applies to very distant galaxies. Or stars receding from us at great speed within our galaxy. There are some very fast moving stars, so fast that they are noticeably red shifted.

So, the red shift doesn’t actually have to make a star redder. It shifts all the light towards the red. But that means it also shifts ultraviolet light into the visible area of the spectrum. So that will become blue.

It’s true that the most distant stars, galaxies and quasars (bright cores of very active galaxies) are moving away from us at great speed because of the expansion of the universe, and they do tend to be redder.

The red dot in the middle of this photograph is a very distant quasar, ULAS J1120+0641. First to be discovered with a red shift of more than 7. And yes it does look very red.

However, that would be a very inaccurate way of testing for the differences. How would you tell the difference between a red giant, and a blue giant that’s moving away from you at a great speed? Or a galaxy that just happens to be redder, or bluer than another one? For instance our galaxy has lots of young stars, and the brightest young stars tend to be blue, so it looks bluer than an elliptical galaxy with lots of old stars in it and not many star forming regions.

The way we notice the red shift is instead because of shifts of lines in the spectrum. These lines are caused by various atoms such as Helium, Hydrogen etc that capture photons and absorb light of particular frequencies preferentially.

This shows the simulated shift of the lines for BAS 11, a super cluster of more than twenty clusters of galaxies, a billion light years away called BAS 11. Left chart shows the sun, right chart shows the distant galaxy. For more about it see Red shift of a distant super cluster of galaxies