So I've heard that all black holes that have been observed rotate to some degree. But if it has zero radius, how is it even possible for the black hole itself to be spinning? Or is it just all the junk inside the event horizon that's rotating? Maybe it spins with infinite angular velocity and a finite angular momentum? Please explain how this works.
Don't think of the singularity as being an object made of matter. A black hole is a vacuum solution to the relativity equations. That means there is nothing inside the black hole.
A black hole doesn't contain matter, but it still has mass. The mass of the black hole can be observed in the curvature of space-time around the black hole. It also has angular momentum, this causes space-time around a spinning black hole to be dragged. This is weird. But this is what relativity predicts, and it's predictions are well supported by observations.
In a black hole, there is no object in the middle, it is the space-time itself that is rotating.
There are multiple solutions to general relativity which allow for multiple different types of black holes. The "normal" black hole you see most people talk about, with a zero-volume, point singularity, is known as the Schwarzschild black hole. If the black hole is spinning, the Schwarzschild solution no longer applies and you're talking about a different type of black hole. This new type is referred to as a Kerr black hole (other types include the Reissner-Nordström and the Kerr-Newman).
In a Kerr black hole, the singularity at the center is still zero-volume, but it is no longer a point. Instead it is a disk of zero height, often referred to as a ring singularity or "Ringularity". The angular momentum of the spinning black hole is then the angular momentum around the axis of rotation passing through this ring.
One way to think of a black hole is that it is what is left behind when some matter (or energy) collapses so far that an event horizon forms. After that, no information of any kind can get out past the event horizon, so what happens inside has no effect on the rest of the universe. The externally visible properties of the black hole (basically the gravitational field outside the event horizon) sustains itself from then on (that's what we mean by saying it's a vacuum solution to the field equations)
If the matter that collapsed had angular momentum, that fact is reflected in the gravitational field, and (among other things) will drag nearby matter around the event horizon. That is what we talk about when we say the black hole is spinning. More precise is to say that its gravitational field is spinning, and still more precise to simply say that the gravitional field matches the Kerr solution to Einstein's equations.
What is actually happening inside the event horizon is (in straight GR, not quantum gravity) not something we can ever know directly. GR lets us predict some of it, but we can never check those predictions and there are things GR is silent about.