Ok that explains how space becomes timelike. But I never hear anyone address how time becomes spacelike. In what sense does it become spacelike? Can one move back and forth in time inside the event horizon?
To answer your question in clear terms: No, you cannot move freely in the radial direction. Only towards the center, in the same way you cannot move freely forward or backwards in time.
Closed orbits are impossible though, all test particles finish at the singularity in a finite amount of proper time.
It is the point at which nothing, no matter how fast or massive, can possibly return from the gravitational pull. For instance: light, traveling at the speed of light in a vacuum, cannot escape once it has crossed the event horizon; it's pulled inevitably inward.
Not quite what you seem to be asking, but the event horizon is the point where[0] the escape velocity is equal to the speed of light; the orbital velocity at that distance is greater (by a factor of ln 2 IIRC, so v_orbital ≈ 1.44c). There's a more distant distance, called the innermost stable orbit or the photon sphere, where the orbital velocity is equal to c (so photons will orbit if you emit them tangentially at this height), but the escape velocity is only ~0.69c.
0: It's not quite right to say that that's because the escape velocity is equal to c, though.
> In what sense does [time] become spacelike? Can one move back and forth in time inside the event horizon?
This can be seen best in a conformal diagram or Penrose diagram (now that he has a Nobel prize might as well use the name of his creator).
Here is the Penrose diagram of a star collapsing gravitationally into a Black Hole and its evaporation via Hawking radiation: https://fias.uni-frankfurt.de/~hossi/Bilder/BR/bhevap_l.jpg
You should track the r=0 line, initially it points upwards (it's timelike) as it chugs along at the center of the star as it collapses gravitationally. When the BH forms, it becomes horizontal (spacelike) and lies at the future of every test particle that enters into the BH. The singularity is inevitable for anything that crosses the horizon.
The Black Hole will at some point have radiated all its energy in Hawking radiation at which point it disappears and r=0 becomes timelike again.
Unfortunately it's hard to make it easier to understand without indulging in some math.
How it happens? Normal spacial direction has positive sign. Time has negative. When crossing the horizon the equations become such that t gets a positive sign and the radial direction a negative one. After that they behave as expected.
Delaying your fall is precisely the same as trying to avoid tomorrow by moving around. Moving away from the center is like moving back in time. No matter how you move in the spatial dimensions you can’t do it.
With the exception of faster than light travel. That allows you to move away from the center, and it also allows you to avoid tomorrow. It’s pretty much the same as time traveling in general relativity.