I'm sure that if the solar system contained a black hole, it'd be a major priority to get a space probe out there to study it firsthand.

But it doesn't; the nearest black holes are hundreds or thousands of light years away, meaning a probe would need to travel at the speed of light (far faster than anything we could build) for hundreds or thousands of years to reach it, and the signal would take just as much time to be sent back. We simply do not have the technology to visit other star systems and other astronomical objects beyond the solar system.

Even if we did, the same physics that make it impossible to detect information about the inside of a black hole with telescopes from the outside, also makes it impossible to receive data from within the event horizon. We'd basically be studying the same phenomena we see around black holes already--gravitational lensing, accretion disks, relativistic jets--but from up close instead of through telescopes.