I just took an astronomy class and it mentioned space was expanding. Light from afar was being redshifted was one of the reasons we know space is expanding apparently.
Light falling into a black hole was also said to be red shifted to oblivion.
So, past the event horizon, is space expanding? I was taught the black hole was a singular point, so I assume space in itself isn't expanding, though.
You are mixing 3 different things:
Time on the outside and near a BH (black hole) is dilating becuase of the huge gravitational field near the horizon. GR (general relativity) solutions like the Schwarzschild solution for the outside of a spherically symmetric body, and more general arguments, show that time dilates Ina stronger gravitational field. In the movie Interstellar that is why the person who was near the BH aged much less that those far away. This has nothing to do with space expansion in the cosmology. The light is redshifted near the BH for the same reason, time goes slower, frequencies go down.
The cosmological expansion is indeed true and would cause a red shift for any light emitted far from you in the universe. But the redshift caused by the BH itself is hugely bigger, dominating any cosmological effect. Similarly, space is expanding between you and the light bulb in you kitchen, but it's so small you won't notice it. Also, as the comment on a possible duplicate by @stafusa will lead you to, spaces pa ding cosmologically has no (or totally negligible effect) on bound matter like plates and knives, or atoms, or planets. The effect is too small, and the forces that keep atoms and planets together easily overwhelm the cosmic expansion. Many trillions or more years from now the expansion will have accelerated and start breaking up galaxy clusters, the galaxies, and so on. No need to worry about it now.
Inside the horizon the BH is not just a singularity point, but it seems to be a region of spacetime separated causally from ours, where the GR solution implies anything falling through the horizon inevitably falls into the singularity. The cosmological expansion has not effect. It actually can be a deeper question because we still don't know what happens insider BH, that will have to await a quantum gravity solution to figure out. But a BH interior is a stationary spacetime, with minor prerturbations just affecting it by slightly changing its BH hair, generally mass, spin and charge. Any light it absorbs from the rest of the universe just changes those slightly if any. Cosmological expansion, to a first, second and third approximation (just to make the point) has no effect.
Well its more like space is falling into the black-hole, and hence the light is red-shifted. Though all these analogies are bad. More technically its due to the time-dilation caused by black hole.
