If black holes warp space-time infinitely, does that mean space-time goes on infinitely in all directions? My thoughts are if a black hole warps space-time infinitely then does that mean space-time goes on in all directions infinitely? 
Maybe (probably) my understanding of space-time is not great, if something can warp space-time infinitely and there is a finite amount of space-time then the infinite warping would pull it all in.
There are a couple other explanations, a black hole is actually a tear in space-time which I have heard before. Or, a black hole is infinitely stretching space-time which means it does not actually need "more space-time"  it just stretches the local space-time infinitely.
 A: Most orthodox theory is that black hole has singularity at it's center. 

A gravitational singularity or spacetime singularity is a location
  where the quantities that are used to measure the gravitational field
  of a celestial body become infinite in a way that does not depend on
  the coordinate system. These quantities are the scalar invariant
  curvatures of spacetime, which includes a measure of the density of
  matter. The laws of normal spacetime could not exist within a
  singularity.

Black hole also has event horizon this is the area in space around singularity where if anything passes it (including light) there is not way for it to go back.

An event horizon is a boundary in spacetime beyond which events cannot
  affect an outside observer.
  One of the best-known examples of an event horizon derives from general relativity's description of a black hole, a celestial object so massive that no nearby matter or radiation can escape its gravitational field. Often, this is described as the boundary within which the black hole's escape velocity is greater than the speed of light. However, a more accurate description is that within this horizon, all lightlike paths (paths that light could take) and hence all paths in the forward light cones of particles within the horizon, are warped so as to fall farther into the hole. Once a particle is inside the horizon, moving into the hole is as inevitable as moving forward in time, and can actually be thought of as equivalent to doing so, depending on the spacetime coordinate system used

Taken from here and here
Space-time does not go in all directions as time always moves forward due to always increasing entropy. It's just that once event horizon is passed the only direction matter or light can go is closer to singularity and never outside event horizon.
A: I think you're mixing warp and stretch, treating them like the same term, and that might be the problem.  Easy to do since the graphic representations of gravity wells seem to stretch spacetime in a 'downward' direction toward the 'bottom' of the gravity well.
https://en.wikipedia.org/wiki/Gravity_well
According to the graphic representation, a warp of more than 90 degrees would start to enlarge or hourglass the bottom of the gravity well. The above answer states, "the gravitational field of a celestial body become infinite in a way that does not depend on the coordinate system," so the maximum or infinite warp is 360 degrees.  Since there is no space to move around in a black hole, all directional momentum is converted into spin, for objects that fall (implies directional momentum) into a black hole.
In this visual idea of a black hole, the lines of spacetime that lead into the gravity well are not connected to the circles of spacetime inside the event horizon, thus the "tear" in spacetime.
A: The way I understand it, is that gravity doesn't effect light directly because light has no mass.  I understand that light follows the curvature of space-time.  Gravity effects space-time, curving space-time more, the more gravity there is, therefore light following that curve of space-time.
If there was a singularity in a black hole, then there would be something for light to "hit" and means that light would follow the curve of space-time back out of the black hole.  We know this doesn't happen.
So what I have understood, is that you are correct in saying black holes bend space-time infinitely, which is why we never see the light come back out, and why it doesn't make sense that there is a singularity.  There can't be a singularity if the black hole curves space-time infinitely.
All that said, light follows the curvature of space-time into a black hole infinitely.  This means that a black hole contains an infinite amount of space-time in all directions (once inside the black hole past the event horizon).
Given there is matter (energy) "falling" into the black hole, where there is an infinite amount of space-time, what do we have now?  Another universe.
So here we are now, the inside of a black hole, inside of a black hole, inside of a black hole... infinitely.
Here's some references that kinda cover it:
https://www.insidescience.org/news/every-black-hole-contains-new-universe
