What processes did Americans that wanted to visit the Soviet Union have to go through during the Cold War? What about Soviets that wished to visit the US?

I'm an average Soviet citizen in the mid-60s with a yearn to travel. How do I go about getting a passport? with answers by u/AyeBraine and u/Dicranurus and u/Kochevnik81 deals with various aspects of the Soviet end of things.

What was it like to cross borders in the Soviet Union? by u/Holokyn-kolokyn talks a bit more about various parts of this idea.