When light changes to one medium to another does it change speed instantaneously or does it slow down/speed up very quickly. Please explain.?

At the speed of light, humans cano lyprerceive it instantaneous

It changes speed in just one molecular layer. So not "instantaneously", but faster than just "very quickly". Basically, the speed changes in a few times 10^-18 seconds.

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You're going to have problems with this "law of continuity" when it comes down to quantum physics. In the real universe that we live in, there are some things that happen instantaneously. Not just in 10^-21 seconds, but actually in an instant. There are continuity conditions in quantum physics, but there are also some instantaneous actions. Such as quantum tunneling inside atoms, or non-local entanglement.

The photons do not change speed. They always scoot along at the speed of light in a vacuum...C.

So why the refraction index n = C/c; where c < C is the speed of light through a medium other than a vacuum?

c is not the speed of the photons when they are moving within the medium. It's the average speed c = S/T where S is the distance within the medium and the total time to make that distance is T.

But here's the deal. That total time includes the travel time when the photons are free to move about and the dwell time when the photons are coupled with the electrons bound to the atoms of the medium.

So if S is through a vacuum, we have C = S/t. But when those photons are captured and then next generation ones are released, that takes time, dwell time, dt. So depending on how often photons couple and decouple we have the total dwell time SUM(dt) that we have to add on to the vacuum travel time to get the total time T = t + SUM(dt) > t.

And that means c = S/T < S/t = C on average.

And this explains why reflected photons and photons that pass on through S exit the medium at exactly C, the speed of light in a vacuum. They never slowed down.

At the speed of light, humans cano lyprerceive it instantaneous

Since it has no mass, it changes immediately. For electromagnetic waves/particles, there is no such thing as acceleration or deceleration.

instant