Kennedy's famous speech on 22 October is a good place to start here:

It shall be the policy of this nation to regard any nuclear missile launched from Cuba against any nation in the Western Hemisphere as an attack by the Soviet Union on the United States, requiring a full retaliatory response upon the Soviet Union.

So he's saying that if any missiles are launced from Cuba, the US response will be to launch everything it has at the USSR - not (just) Cuba. The USSR would detect these launches and launch its own strikes.

Neither side would have a lot of time to carefully select targets; they would be operating according to set plans, so for instance NATO bases in Western Europe and Warsaw Pact bases in Western Europe would also be targeted. By the time that hundreds of massive atomic bombs had gone off - well, nobody really knows what would happen and nobody really wants to find out, but assumptions were that it would be pretty bad.

This is the basis of MAD, of course - "Mutually Assured Destruction". The concept that it must always be impossible for a nuclear war to have a "winner", because if one or other power felt it could start one and win it, then at some point somebody would do just that.

Now, the main fear during the Cuban Crisis was actually less that one of the missiles on Cuba would get launched at the USA, but that the USA would trigger war with the USSR by taking direct military action against either the installations by bombing them from the air, or against the additional missiles being shipped from the USSR.

As anyone who's seen "X-Men: First Class" knows, a line on the map was drawn and a Soviet freighter carrying missiles was heading directly for it (though I don't believe in real life it was accompanied by a Soviet surface fleet - that would kind of defeat the whole "plausible deniability" thing of using a civilian theatre in the first place). If the US made good on its promise and sank the freighter, they'd be directly killing Soviet citizens.

If that happened, although it was never made explicit, it can be reasonably assumed that in order not to look weak in turn, the USSR would have to undertake some kind of retaliation, which would probably escalate into a full-scale war.