It depends on the period in time. Early in the Cold War, well into the early 1960s, the US doctrine was very rigid. You used all nuclear weapons or none of them. There were no options. Part of this was institutional and part of this was technological: the weapons themselves were vulnerable to sneak attacks so it was seen as a "use it or lose it" situation.

By the later 1960s and especially early 1970s the war plans got more diversified. There was something of a "menu" of options that the Commander in Chief could order, ranging from "nuke 'em all" to "only nuke a few things and see what would happen." But most planners and policymakers assumed it would escalate to "nuke 'em all" rather quickly. Why? Because there would be almost no time to react — the time between detecting an incoming attack and retaliating had gotten very, very short — and there would likely be no reliable communication with the enemy. (Even the famous "hot line" that Kennedy had installed after the Cuban Missile Crisis was only a civilian-grade line, not intended to survive any kind of nuclear war conditions.)

The US didn't believe that it would necessarily just be a one-off, all-out, "we'll-all-go-together-when-we-go" attack. They planned to fight limited nuclear wars and even strategic nuclear wars over the course of several weeks. They found in many of their simulations of this that things went pretty badly, that even under optimistic conditions it became pretty hard to remain a cohesive fighting nation state if the other guy was nuking you on a regular basis.

If you want an idea of what a nuclear war plan looks like, this is the only official "map" I have found detailing one. It comes from a 1958 Strategic Air Command film illustrating how they would fight a nuclear war. It isn't necessarily an actual war plan but it has the flavor of one. (It looks strange because it is a composite map made from many screenshots.) It is a little hard to read (it is a gnomonic projection which is not familiar to most people, and the quality is not ideal), but the North Pole is mid-way along that thick black line, the US is in the lower right corner, and the USSR and China are outlined as targets at the top left, Spain and North Africa are at upper right.

It does a good job of showing how the use of overseas SAC bases and aerial refueling was meant to allow the US bombers to penetrate very deep into the USSR. Each of those receding "border" lines show the penetration of planes 1 hour after crossing into enemy airspace. Note this was still in the era where the US could literally not target China as well, even if China had nothing to do with whatever attack was going on. Of note is the plan to refuel at one of the airbases near Tokyo, and other places that were ostensibly non-nuclear but were part of the overall logistics of attack.

Anyway, you can see that they were planning to send a LOT of planes at the USSR and China, to pretty much every major population center, industrial center, and military center. They were planning to launch bombers from bases all over the world. The fallout from such an attack alone would contaminate any areas not hit by the blast.

Similarly the US assumed the USSR was going to nuke them pretty hard. Here is an assessment of fallout risk from a Soviet attack in the 1980s, created by Oak Ridge National Laboratory. They were assuming the Soviets would target both population centers and military bases. That's a lot of radiation, incidentally, that they were predicting.

So the long and short of it is, at different times the US and the Soviets thought they could fight a "limited" nuclear war. But the planners at the top of both of them more or less assumed that it would end up quickly being a full nuclear exchange no matter what anyone wanted. The military was prepared for that as well.