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How can a photon have no mass and still travel at the speed of light?
If light travels at the speed of light, and anything with rest mass will experience relativistic effects based on the Lorentzian equations, why doesn't light experience these kinds of effects?
For example, relativistic mass and rest mass are related via
$$m = \frac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}}$$
Shouldn't light therefore have no rest mass (since the Lorentzian is $0$)?
You are quite correct, light has zero invarient mass.
Light does experience relativistic effects, but because $\gamma$ goes to infinity at $c$ you'll get confused using the Lorentz transformation to try and understand what is going on. The best way to understand the effect of relativity on light is to note that light travels on null geodesic. We can use the heavy machinery of general relativity to calculate the null geodesics for any spacetime and hence work out what happens to light. For example this allows us to calculate the bending of light in a gravitational field and also the fact light can't escape from a black hole.
