Do objects' velocities slow down going into black holes? Would they appear to? I always hear that time slows down as you enter a black hole, and when I hear this, I always mentally imagine all of a person's motions, including their net velocity relative to the black hole, slowing down, but I'm unsure of the latter part. Would a person's velocity going into a black hole, as seen by some far away observer, be fast or slow?
I'm pretty sure that a person's velocity would appear to slow down, based on the effect of photons having to traverse more and more spacetime as the spacetime curvature near the black hole gets extreme, but if you were a good observer and calculated these corrections, I want to say that the speed of entry that the observer would calculate would be quite fast. This would correspond to the intuitive thought that the intense gravity pulling on the person should make them accelerate as they go into the black hole. 
Edit: Put another way, I understand that, from the observer's point of view, an object falling into a black hole falls in within a finite amount of time. Would this time correspond to


*

*The object approaching the black hole at constant speed

*The object approaching the black hole at increasing speed

*The object approaching the black hole at decreasing speed


I'm not experienced in GR, so I'm probably trampling over all sorts of subtle concepts, but any help in sorting these effects out would be appreciated :)
 A: 
pretty sure that a person's velocity would appear to slow down

It would slow down in the sense that you would get bored watching the images of the person move slowly and redly as it traced out a finite amount of proper times actions over what seems like forever.

based on the effect of photons having to traverse more and more spacetime as the spacetime curvature near the black hole gets extreme

An event horizon is not a region of extreme event horizon. And event horizon could form inside you at any moment and expand outwards and it might be a trillions of years before anyone notices that you were the center of a newly forming black hole.
An event horizon is a cutoff between when you can escape and when you can't. As far as distant observers notice it is just another part of the $t=\infty$ hypersurface, just something you never see things reach.

but if you were a good observer and calculated these corrections, I want to say that the speed of entry that the observer would calculate would be quite fast.

Speed relative to what? You could measure the radar distance and radar time of the events of the object approaching the black hole. But this is just a coordinate speed a difference in one coordinate over a difference in another coordinate. It isn't very physically relevant even though radar time has a fine operational definition for events outside the horizon.

This would correspond to the intuitive thought that the intense gravity pulling on the person should make them accelerate as they go into the black hole.

Now you have a good question. In the weak field Newtonian limit it looks like you can describe things with acceleration. But should this be change in momentum per unit time or should it be change in momentum per unit proper time. These are different concepts entirely, but they have the same Newtonian limit. This is a general problem. There are lots of things that are different but have the same Newtonian limit. For instance ma and dp/dt have the same Newtonian limit but they are different. So you really should throw away Newtonian physics learn it right, and then see how to do the Newtonian limit, and then you'll see which was the right idea.
In this case, you can even consider a beam of light heading straight in. It isn't going to fall any faster. But the horizon is an infinite radar distance away as well as an infinite radar distance away. Which is perfectly reasonable given that anything you are seeing heading in could still come out.

Put another way, I understand that, from the observer's point of view, an object falling into a black hole falls in within a finite amount of time. 

Only the object that crosses the horizon (or some others that cross) think it took a finite amount of time. The outside observers don't even know whether it actually crossed or turned back.

Would this time correspond to
The object approaching the black hole at constant speed

For the infalling light ray and using radar time and distance for the outside observer you get that it moves at the speed of light. And since it is a coordinate speed that shouldn't be discounted too easily. But it does mean that the other objects go less than light speed (and again given that space can expand that shouldn't be discounted too easily either). But here is an example of the particle falling in at constant (light)speed. And it take infinite radar time and infinite Schwarzschild time to get there.
Since it takes infinite time for an object rushing inwards at the speed of light, it takes infinite time for slower moving particles to cross it.
