The Cuban Missile Crisis is probably the closest humanity ever got to destroying ourselves and the world intentionally. While we know the contributing factors now, how would the average citizen of the nations involved be responding to the event as it happened? What did reporting on it look like, and how did it vary between each country? Were people packing their things and desperately fleeing to fallout shelters, shuttering themselves in their homes with the news on just holding their breath, or just going about their daily lives? How much information was each government letting through to their citizens? The pipeline for information in 1962 was not nearly as streamlined as it is now, so are there any recorded accounts from people who just completely missed this and freaked out when they heard?
I know that's a lot of questions, thank you all in advance!
I can only barely scratch the surface, because as you say that is a lot of questions, but I can talk a little about how the crisis was perceived in each country. On the topic of what people actually did in response, whether that was hunker down in their backyard shelters or just continuing as normal, you may have to wait for someone else or ask again.
What I will say for sure, though, is that your intuition is spot-on: the government played a very important role in shaping public perception, not necessarily by completely restricting information, but by wrapping that information into a narrative. So let's look at those narratives, country by country.
In the United States, it's the "Cuban Missile Crisis". You probably already know about how that limits understanding geographically. The idea of the crisis being "Cuban" already focuses the narrative on the western hemisphere and on the US, and it cuts out the fact that the United States had missiles at a similar remove from Moscow in Turkey. This narrative frames the crisis as a relatively short blip; "thirteen days", according to Robert F. Kennedy. It started when the US found itself threatened by missiles in Cuba, and ended with an assurance that no more missiles would be put on the island. It's a very US-centric framing, of course.
What I'm ultimately getting at, though, is that these facts — none of which are untrue — were selected as relevant by the Kennedy administration and thus by the press because they conformed to a rhetorical image. Missiles in Cuba threatened Washington physically, sure, but also rhetorically. They threatened its claim to be defender of the "free world" and to a masculine, upright posture in foreign affairs. If you are a US citizen in the Cuban Missile Crisis, much of what is going through your head is fear, sure, but it's fear induced by the fact that your self-image as an American is threatened.
In the USSR, it's the "Caribbean Crisis". That's a subtle shift, but it's important. The missile incident is just a small part of a much larger succession of events, all of which add up to a consistent pattern of the US trying to subvert and harm this fledgling socialist state that the USSR, as leader of the socialist world, has pledged to protect. Again, this is a particular framing of events that emerges from, and reinforces, an identity, that of the USSR as protector of the recently-independent and threatened. If you are a citizen of the USSR, you probably believe that your country took a reasonable step to defend Cuba from American aggression, which did not spark a crisis so much as inflame American rhetoric in an existing, longer-term crisis. This is why Khrushchev was satisfied with a promise from the US not to invade — it strengthened the USSR's identity as protector.
In Cuba, meanwhile, it's the "October Crisis". This narrows the chronological framing back down, but it also broadens the importance of the events within that shorter period. After all, in Cuba, the crisis is not happening on your periphery — crisis is your experience of October 1962. In the Cuban framing, greater superpower conflict is not the point. The crisis was the threat to Cuban sovereignty and independence posed by the violent, aggressive behavior of the United States, of which the Bay of Pigs invasion had been an earlier example. Again, the crisis was framed in a certain way based on the identity of the people doing the framing. Although I said I wouldn't talk too much about whether people fled to bomb shelters or went on with their lives, I'll break that here just to say that in Cuba, people went on with their lives. Another day, another attempt by the US to reassert its control over them.
Sources:
Weldes, Jutta. "The Cultural Production of Crises: U.S. Identity and Missiles in Cuba." In Cultures of Insecurity: States, Communities, and the Production of Danger, edited by Jutta Weldes et al., 35–62. Minneapolis: University of Minnesota Press, 1999.
FitzGerald, Réachbha. "Historians and the Cuban Missile Crisis: the Evidence-Interpretation Relationship as Seen Through Differing Interpretations of the Crisis Settlement." Irish Studies in International Affairs 18 (2007): 191-203.