One can break it into rough phases. First, during the Stalin period, there was essentially no public discussion of nuclear weapons. The official position of the Soviet state was along the lines of: nuclear weapons don't matter that much, they don't fundamentally change things, they're not even that hard to make, don't stress over them, we aren't. In private, of course, after Hiroshima they were racing to acquire them, but there was almost no public discussion of them except to point out that they didn't give the US some gigantic advantage (which wasn't totally wrong in the early days of the US nuclear monopoly, when its arsenal was quite modest and difficult to deploy). As a result of this approach there were basically no serious attempts to prepare to lessen the impact of an attack (e.g., blast and fallout shelters).

If we jump ahead, though, to the post-Stalin period, especially under Khrushchev, you see a big shift. Now the USSR has many nukes (but not too many that can hit the USA directly), the USA has lots of them and can deploy them easily against the USSR. Khrushchev saw nukes as a form of state power and proudly paraded his missiles through Red Square, announced the testing of huge bombs, and nearly started/narrowly avoided nuclear war through the deployments in Cuba, and so on. Nuclear weapons became a big part of Soviet public consciousness, and so did preparations for their possible use. The USSR started large, well-advertised projects for public shelters — blast shelters, mainly, but also fallout shelters. They started training people for Civil Defense (things you can do if the bombs start dropping). So somewhat similar to what was being done in the US during this time, albeit with a lot more state sponsorship (the US push for fallout shelters in the early 1960s was about building your own fallout shelter, not about the government making public ones, on the whole).

The Soviet approach to Civil Defense was more internally funded and broader than the US one, but it still was pretty inadequate given the size of the problem. The Soviet leadership was under no illusions that it would save their population or industrial base in the case of nuclear war, or that a total nuclear war was "winnable." I bring this up because there is a lot of Cold War-era US coverage of Soviet Civil Defense which emphasized how big it was compared to the US and how that meant the Soviets believed they could survive nuclear war and would be more belligerent and so on. All of that seems rather overblown, according to Soviet historical records.

As in the US, you eventually start to see a souring of Soviet public opinion towards these kinds of things — a disbelief that a subway tunnel or hiding under a desk or a few feet of packed dirt are going to protect you against a thermonuclear warhead (much less 10 or so; the US arsenal in this period was huge and redundantly aimed). Which both is and isn't correct (the idea that such measures would guarantee survival is false, even if they were deployed more widely than they were; the idea that they are pointless or wouldn't make a large statistical difference is also probably false — saving an extra 10-20% of the population is not nothing!). They had grim jokes about this stuff just like in the US by the early 1980s:

"Comrade, what should we do if the Americans launch a nuclear attack?" [a worker asks the party secretary.] "Simple," the party secretary replies. "You wrap yourself in a sheet and make your way slowly to the cemetery." "Why slowly, comrade?" the worker asks. "Simple again," the party secretary replies. "So as not to cause panic."

However unlike the US, they didn't just sort of abandon all of this preparation in the 1970s, and still have courses on Civil Defense in many Russian high schools, apparently, to this day (so my students from Russia have told me; they say it is sort of a mixture of "survival" and "home economics").

Anyway, there is a lot more that could be said, but this is just a very broad overview. The best book on this subject is Edward Geist, Civil Defense in the United States and Soviet Union, 1945–1991 (UNC Press, 2019), which is a comparative history of the Soviet and US approaches to Civil Defense.