Did a regular Joe in the late 70s know about stuff like the Minuteman, SLBMs, nuclear test facilities, and the nuclear triad?

There's no reason they wouldn't. These were openly discussed in Congressional testimony and newspaper articles. The basic aspects of US nuclear deterrence capabilities and goals was not a secret, even in the early Cold War. It was part of the US strategy to talk about these things; it made them credible, and it made the public a participant in the spending.

How much did they know compared to what we know now?

It depends on what you mean "know." I don't mean to be pedantic about this. Did they know that Minuteman missiles were solid-fueled ICBMs at specific bases? Yes. Did they know the exact Circular Error Probable of their accuracy? Probably not, but there existed enough information to estimate it. Did they know their exact yields? I don't think so. Did they have a good sense of what the yield could be? Yes.

"Open-source intelligence," of the sort that now populates Wikipedia and the like, was only just starting in the late 1970s. These are the professional estimates of classified information by experts, culled from lots of public testimony/reports with independent analysis added. It rose to something similar to what we have today in the 1980s. So some things were known and available, some things were not.

Let me give you an example of this. "How does an H-bomb work?" is the sort of thing that, prior to 1979, there was speculation about, but nothing really "concrete." After 1979, the now-familiar diagram of the two-stage Teller-Ulam diagram was ubiquitous. We can trace that particular release to a very specific event (the U.S. vs. Progressive case, 1979). Did people prior to this case have no idea of how it worked? No — one can find the basic idea well-represented even in open materials from the 1960s, and versions of that material were useful to the activist at the center of the case in 1979. But there were a lot of confused and competing ideas around at the same time. It wasn't until the case that one particular version of the information became "validated" (by the government trying, and failing, to censor it), and thus it became (to some degree) truly "known." Knowledge isn't a binary thing (you know it, you don't), it's a more complicated process, with many shades of grey. (One could argue that at some level we still don't "know" many of these details.)

For any given question about nuclear capabilities you can apply a similar historical (even genealogical) approach to its knowledge and its public awareness. But for "high-level" concepts, like the nuclear triad, it's relatively easy to say when they started to get deployed, because these are the ways in which the military talks to the public. (The "nuclear triad" idea only emerged when the military was trying to defend the need to maintain three separate means of deterrence — the concept was developed as a defense against budget cuts more than as an actual strategic concept, and is still deployed din this fashion.)

How much might an enterprising and interested researcher learn without security clearance?

One starts to get what I call (in my book) "secret seekers" in the late 1970s, but it was still a somewhat difficult job. Congressional testimony contained a lot of information, though key numbers and concepts were sometimes censored out. There were organizations at this time, notably SIPRI and the NRDC, that employed researchers to go through such open-source information and synthesize it. But that activity, again, really takes off in the 1980s.

If so, how technically detailed was this information compared to "there are missiles and they go boom"?

By the 1970s there was a pretty robust public understanding of different types of missiles, different types of warheads, different types of capabilities. Again, it depends what we mean by "knowledge." Did they know exactly what kind of warhead design was used in a Minuteman missile? Probably not. Did they know many key pieces of information about how it would work, what its accuracy and response time was, what its yield was? Sure, to some degree of "known."

(Bonus question if you were willing to watch the link) If I were a citizen of 1977 America and happened to somehow have a script of CGP Grey's Tekoi video, and we skip the part about me being a time traveler and/or crazy, would I go to jail for publishing that in a newspaper? Which parts would have been sensitive at the time?

I haven't watched the video but I can say that in terms of publication, the key legal question is whether the information came from an official source (a genuine leak of classified information) or whether it is speculation/inference/guessing/entirely open-source research. In the former case, whomever leaked the information could definitely go to jail, and there were cases of that from around that time, as happened with Samuel Loring Morison in 1984, when classified information he leaked to Jane's Defence Weekly was published. Morison is the only one who has ever been convicted of this, though; it's a rare thing. At a minimum, however, if someone starting writing about things that showed a knowledge of definitely classified information, it would trigger an investigation into how that information got out.

In the latter case, where the information could be traced to open-source analysis, the odds of a prosecution are essentially zero. And the journalist writing the article, if they are just an intermediary, would be unlikely to be prosecuted. In general it is very difficult to prosecute journalists for such things in the United States, because of the First Amendment.