The largest Communist Party organization in the United States was the CPUSA. Under pseudonyms, they held secret meetings throughout the United States that represented the skeleton of the Communist Party in the US. Your average radical-thinking US citizen would need to somehow navigate his acquaintances (I would imagine there existed strategies for this) to find someone that knew something about local meetings. During the Second Red Scare, the time period you are describing, this would have been very risky business for one's reputation. Once you were able to find a meeting, however, your ability to meet well-connected people, eventually leading to the Soviet Union itself, would be exponentiated.
You make a contact with the Soviet embassy and you say you want to become a spy. If you're just an average American you'll probably not have much luck, but if you can prove your connections and you have valuable data then you have a reasonable chance of being recruited.
That's how Aldrich Ames and Robert Hanssen did it, although they are probably special cases since they worked in counter-intelligence and thus had knowledge of the leadership of the Soviet/Russian spy organization. However, it's also how John Walker became a spy and he was merely a naval officer.
But that's more or less the most direct route if your decision to become a spy came out of the blue. If, on the other hand, you had a long history of Communist affiliations and ideological leanings then it's possible you could become a spy through different, more "organic" means. There were various Soviet agents within many Communist party organizations in the US, often looking to groom folks for spywork if they had valuable connections (politics, military, research, etc.)
I'm not a historian, but I am a political science graduate student and the recent history of communism and the history of nuclear weapons are something that interests me.
What do nuclear weapons have to do with your question? Well, I recently read Richard Rhodes's magnificent books on the development of the atomic and hydrogen bombs. The first book, The Making of the Atomic Bomb focuses entirely on the west, and eventually the US. In the second book, Dark Sun: The Making of the Hydrogen Bomb, he backs up a little bit and gives us the history of the Soviet atomic bomb project. As part of that history he gives a fascinating account of how Soviet spying worked in the West, particular the UK, Canada, and the US in the 1940s, giving particular attention to a man named Harry Gold, who became Klaus Fuchs's contact in the US.
Basically (as Rhodes describes it) you'd join the local communist party, which would include Soviet agents that would look for individuals with particularly useful affiliations. These groups would effect an atmosphere of conspiracy by their nature, something the recruiting agents would foster. The meetings would be held in secret, plans would be secret, etc. (Rhodes doesn't describe it this way, but I read it as being useful as subterfuge against local authorities as well — "the commies are meeting in secret".) Once an individual is recruited, they drop out of the party "cell" and are no longer involved in party activity, isolating him (or her) from the other members.
The other interesting thing about Rhodes's narrative is the revelation that Soviet espionage (at least on the atomic bomb) wasn't one-way — the Soviet scientist in charge, Ivan Kurchatov, when shown the intelligence gathered from the west would send back specific queries regarding developments in the US, etc. These would filter down through Gold's contact to him, and he would then try to get the information from Fuchs (Gold was, IIRC, a chemical engineer, and understood something of what was being discussed, although Fuchs was dismissive of his knowledge, according to Rhodes).
Anyway, I hope that helps, and I hope I haven't violated the rules. Others here probably have better sources, but the first several chapters of Dark Sun are particularly informative regarding the Soviet espionage of the Manhattan Project and how Soviet recruiting and espionage worked generally immediately before the Cold War.