Should just appear. Do you have the right quest selected?

I found a black hole by accident. Just jumped into the system. I wasn't sure what t do with it... so I just explored the planets and jumped out. Do I need to go back to it?

Black holes will warp you hundreds of thousands (sometimes well over 1 million) light years around the galaxy. You'll end up close to the center, but the cast majority of that distance is just a big arc to the other side of the galaxy. You can choose to go through them or not. Be warned if you do, one of your ship's installed tech will be broken on the other side and need repairs. This only applies to tech installed in the regular inventory slots, not the tech slots.You can teleport back to any bases you own or space station you've visited via a teleporter. Is that line the path to the center? If Nada (or Polo now, I think, since the Beyond update) reveals a black hole to you, it's usually very close. Should be within a jump or two of your position. You can't see black holes in general on the galactic map until you've completed the Atlas Path quest. Nada/Polo will reveal one, and you can stumble across them accidentally by warping into a system that contains one, like MrSnide did.

NewYears1978: Hopefully you mean from Polo (provided you're playing the Beyond version and not an older version). Otherwise you might have done it wrong. Make sure your filter is set to "Black Hole" on your galaxy map. The right and left arrow keys will change filters, but typically not until you've been in the galaxy map for like ten seconds (after it finishes the whole rotating around the current system song and dance).

TerriblePurpose: I'm not sure if this is still the case. I went through a black hole yesterday and didn't see any broken tech on my ship, and I've been doing a lot of upgrading so I feel like I would have noticed. (I do have several pieces of tech in my regular ship inventory slots instead of in the tech slots, so that's not it)

I'd also concur that the black hole should only be a couple jumps away at most from your current position. I'm pretty sure it put a black-and-white star icon on the system with the black hole in the galaxy map. Note that (at least for me) it didn't put any such marker on the compass up top when I was in the black hole system to show where it was, though. I had to look around to find it visually, though it was within a couple seconds' warp from the station in case that helps. I wonder if NewYears1978 has actually been in the right system but just not spotted the black hole, though.

I haven't traveled through one in Beyond, so you could well be right; HG may have changed this.

No, just tried and the black hole broke the emeril drive.
But it seems there is some basic tech (launch thruster, photon cannon, pulse, hyperdrive) that does not become broken.