After all, it does change direction when reflection occurs. So shouldn't it also accelerate? And since the acceleration cannot increase the speed of light, mustn't it slow down?
Light does not slow down during a reflection.
Light is a signal disturbance in electric and magnetic fields. These disturbances propagate through space at a fixed speed $c$ in vacuum. The situation is completely analogous, in a mathematical sense, to a wave pulse that is sent along a string. When the pulse encounters a boundary, it flips direction, and may or may not change phase depending on the type of boundary encountered. For good graphical depictions of this phenomenon, visit this page.
If you emit a pulse of light at a distance of 1 meter from a plane mirror, and measure the amount of time it takes for the signal to return, you will find that it is 2 meters / $c$, neglecting refractive effects of the air. In this sense, we say that the light has not slowed down, even though it has changed direction in the middle of its journey.
There are some processes which may be termed acceleration. For example a photon having velocity $c$ in vacuum and then having velocity $c/n$ after entering a medium with refractive index $n$. Or as you describe, a change from velocity $c$ to $-c$ when reflected from a mirror. However, the magnitude of it's velocity is $c$ at all times, and in fact in when changing medium the photon is absorbed and a new one emitted, so whether that is to be called an actual acceleration is debatable. This queston is related and may be of interest.
The actual thing that happens, in a more general way than just reflection, is that when the space properties change, for example, a mirror, the particles of the mirror absorb the photons, and emit new ones, the light doesn't reflect but dissapears giving energy to the atoms that then loose it again as a new photon. In an interface of materials, when light passes from $c/n_1$ to $c/n_2$, the same thing happens, the light is absorbed and re-emited by all the particles, and it actually travels at speed $c$ between them, the average behaviour of the wave front is that it travels a little bit slower, when actually the light is not travelling directly in the direction of the wave front, but average.