How far away is the light that would reveal the Big Bang?

I guess theoretically if we could go faster than light, which we clearly cannot, at some point we'd be able to see the big bang itself.

If we travel instantaneously across the universe (as measured by cosmological time), then no, theoretically that's not the case. Our theoretical assumptions involve large-scale homogeneity and isotropy, meaning that the view different observers is basically the same regardless of where they are, at least at the same cosmological epoch. This is called the cosmological principle.

Mind, because of the way relativity works, an FTL drive might also be capable of taking you back in time, so perhaps you could theoretically see the Big Bang after all. But that's obviously more to do with time travel than going to distant places.

Seems like the moment something happened, light was dispatched and unless particles traveled faster than the speed of light to obstruct that light, if I were to appear in front of that light, for a single tiny moment, I'd see that little white dot that would've appeared the moment the big bang began (and then see everything that happened afterward). Why not? Light point A -> same light at point B [observer]. Where's the complication?

You seem to have an implicit assumption about where the Big Bang occurred. It actually occurred everywhere. Including here. See this question.

Imagine looking out to the most distant galaxy that you can barely see. Because of the finite speed of light, you are looking at it not as it is, but as it was in the distant past. You are just now catching light from that distant place.

If you were instantly transported to that place, and look back to where the Milky Way should be, you won't see it, but rather as this area of space was in the distant past. You'll be just catching light from somewhere the Milky Way area.

In other words, you will see light from the distant past no matter where you go. Because the Big Bang happened everywhere, light from it (well, actually from recombination epoch later, because the universe wasn't transparent before then) is present everywhere. And hence it'll look about the same no matter where you go, again assuming isotropy and homogeneity.