How does different colours of light come about? Does red light means the photons are red in colour?

Colour is how human eyes perceive the different wavelengths.  It isn't a real intrinsic property at all.

To ask why the lights have different colours you would need to speak to the designer of humans.

Red light doesn't travel faster in a vacuum.  A material such as glass slows down light through its interaction and it just so happens that it has less of an interaction with low frequency light ( red) compared to the other colours. Note that the intermediate colours a slowed down more than red but less than blue.

As far as HOW we perceive different colours we have special organs called "cones" that respond to the different wavelengths of light.
In some cases there is a deficiency in some of these organs and those people become "colour blind"
i.e they can't distinguish between a pair of colours for example red- green.

White light is the light that comes from the sun. It undergoes something called refraction due to the water molecules in the air as well as some other factors. When white light undergoes refraction it splits in to the 7 colors of the rainbow, Red, Orange, Yellow, Green, Blue, Indigo and Violet. The color of the light is determined by it's frequency solely. These frequencies are again as a derivative of refraction.
This may help explain the refractive process more.
http://www.tedmontgomery.com/bblovrvw/emails/lightfrequency.html

Please note that this light can also be refracted through any medium not just water molecules or pollutants in the air

No, photons are not red in colour and red light doesn't travel faster than other colours of light.
You can imagine light as photons but also as waves: all of them have the same velocity, but different frequency, i.e. the tops of waves are more or less distant each other.
If their distance is between 400 and 700 nm, humans perceive light, otherwise it is invisible.
There could be different reasons why light have different colours.
The first example is that of a light emitter which emits light of different frequencies, if we perceive it as red, it is because among all frequencies red ones are dominant.
In our eye we have cones sensitive to different wavelengths (red-green-blue).
The second example is the one of a white light (comprising all frequencies) being reflected by a red surface. We see it as red because among all frequencies, only the red ones are not absorbed by the surface.