Can a photon have little to no energy and/or speed? Can a photon move more slowly than the speed of light and behave 'non-relativistically,' so to speak.  Perhaps another way to express my thought is: could we stop a photon from moving?
 A: People are addressing the speed question, but just to be clear: a photon can be very low energy. For instance, radio waves are much lower energy than gamma rays, even though both are made of photons (and, in vacuum, both travel at the speed $c$). What determines the energy of a photon is the frequency of the excitation (frequency of the corresponding light wave), which is related to the energy by Planck's constant, $E = h f$. This doesn't affect its speed, and it doesn't really have anything to do with "relativistic" concerns--photons themselves are not necessarily present in relativity. 
Usually relativstic/nonrelativistic is the kind of distinction you get to make with massive particles, since you can talk about their energy compared to $m c^2$. If they have low kinetic energy it will be roughly $T \approx \frac{1}{2} m v^2$. Since a photon NEVER has such an energy, being massless, this doesn't really come up much. Of course, since we usually care about photons only when they interact with massive particles, relativistic concerns can pop up if the photon energy is large compared to the mass of the particle in question (see Compton scattering for instance). It's just never so clear-cut as "oh, this is an infrared photon, let's treat it classically always."
A: Refractive Index is when light travels more slowly in a medium.
Here is an example of light being slowed down to 38 miles per hour.
The speed of a photon does not affect its energy.
It has zero mass, therefore zero kinetic energy.
The energy it has is due to its frequency (color), and nothing else.
(However, it does have momentum!)
A: Although it has been said in other comments and answers, it bears repeating succinctly: photons (as far as any experiment can tell) are massless and therefore always move at the universal, invariant speed of light. There is NO non-relativistic description of the photon. Even the "classical" description of light - Maxwell's equations - can be interpreted as the propagation equation for a lone photon - aka a one-particle state of the electromagnetic field, and you may know that Maxwell's equations are Lorentz invariant: they are fully relativistic equations and were always so as soon as Maxwell wrote them down - indeed historically they were the first model of a relativistic description, whence people derived special relativity. So there is no nonrelativistic classical description of light either.
When we see "light" in a medium, we are seeing a quantum superposition of excitations both of the electromagnetic field and of excited atom / molecule states. So it's really "photons" together with "other stuff" and the photon part still moves at $c$ exactly. The full superposition can move at speeds less than $c$, which means it also has a rest mass. Indeed, the "particles" of an optical frequency disturbance in a glass of index 1.5 have a mass of the order of $10^{-6}$ electron masses, as I calculate here.
Also, as Daniel Sank points out, there is an equivalent classical description of the superposition - hybrid modes - and, when you think of Maxwell's equations as the propagation equation of the one particle EM field state, the mathematics of the two descriptions is identical.
A: I will reply to this because the checked answer is not answering the question.The question is about photons, the answer is about light. It is as if the question were about atoms and the answer is about density of material. The question is asked about photons, the quantum mechanical framework is relevant to it. The checked answer is about light which is in the classical electrodynamics framework. 
Photons are elementary particles with very specific attributes, one of which is zero mass and always moving in vacuum with velocity c, the velocity of light. Light is built up out of zillions of photons in confluence. .
As also WetSavannaAnimal explains in his answer the two frameworks can be rigorously connected mathematically but still photons and light are two distinct physical manifestations each belonging to a different physics framework, and the classical emerges from the quantum underlayer in a mathematically demonstrable way.
To the question:

Can a photon move more slowly than the speed of light and behave 'non-relativistically,' so to speak. Perhaps another way to express my thought is: could we stop a photon from moving?

No, a photon can never move more slowly than the speed of light in vacuum, because as an elementary particle it is either in vacuum or interacting with another elementary particle or field. Its interactions can be elastic, inelastic or total absorption by kicking an electron of an atom to a higher energy level. You can only stop a photon by total absorption/disappearance.
It is the synergy of zillions of photons with the molecules of the specially prepared medium  that produces the effect of slowing light velocity in a medium referred in the checked   answer.
