What happens to an objects velocity as it enters and travels through a Black Hole? Assuming an object is traveling at some speed that is less then the speed of light, and it then enters a Black Hole. Would that objects velocity increase, decrease, or remain constant between the time it passes the Schwarzschild radius and before it reaches the singularity?  
 A: For the following I will consider a static, uncharged Schwarzschild black hole.
We can use GR perfectly well beyond the horizon ($r=2M$) of a black hole. The horizon is just a coordinate singularity of the Schwarzschild metric: it is no physical singularity. It is true that beyond the horizon there are no geodesics out of it but we can still use GR inside. We can not receive signals/"see" the object beyond the horizon (from outside) but that does not mean that there is some crazy stuff happening there.
For discussing motion across the horizon Gullstrand–Painlevé coordinates (PG) are very well suited. These coordinates have no singularity at $r=2M$ the singularity of the PG coordinates is at the center ´($r=0$) of the black hole and this is the physical singularity of the black hole.
The speed inside the horizon stays finite and below the speed of light. Only at the singularity at the center the speed becomes infinte but at that point very close to the singularity GR may not hold on its own since effects of quantum mechanics become important. To begin with I would highly recommend reading the sections "Motion of raindrop" and "Speeds of light" in the Wikipedia article of the PG metric.
In short; the speed of the free falling object increases but stays finite and below $c$ until it reaches the singularity. The acceleration and speed is continuous across the horizon, since again the horizon is only a coordinate singularity not a physical one.
Apart from this it is always a bit difficult to discuss such things in words, since time, distances and speeds are relative: especially around black holes. So the question from where we are observing and describing what is always important and tricky. It is very hard to describe those effects/motions on point/correct (at least for me).
For rotating/charged black holes there might be similar discussions but for those cases I do not have references, so I only discussed the situation for the Schwarzschild black hole.
I am certainly not an expert on geodesics and effects around black holes but I am pretty confident that the bit that I said here should be correct.
