I'll answer this for East Germany, which had a similar, although not exactly the same, system, to that of the Soviet Union. I'm sorry this doesn't directly answer your question, but I'm not nearly as confident in that regard.

The tl;dr version is that you basically couldn't advance very far in East German society without being a party member.

By the time the German Democratic Republic (DDR) really began moving away from direct Soviet control in the early 1960s, the Sozialistische Einheitspartei Deutschlands was firmly established as THE party. Other parties existed in the DDR, surprisingly, but were subservient to the SED and at least partially existed to create the illusion of actual democracy. Those other parties did not have the same permissions as the SED, though, and were just kind of "there."

Communist ideology began at an early age, in the DDR, USSR, and the rest of the Soviet Bloc. Communist thought was heavily promoted in schools, of course, but the Communist Parties (such as the SED) generally had their own youth organizations. In East Germany, this was called the "Free German Youth (FDJ), and in the Soviet Union it was the Komsomol. Participation in these organizations was never really mandatory, but anybody looking to do anything significant with their lives was encouraged to join. You could join in grade school, and could stay in into early adulthood (21 ish), at which point you had the option of joining the communist party. In both the DDR and USSR, soldiers (conscription was very much a thing) were permitted to wear their Komosomol/FDJ membership badges on their uniforms (and, in East Germany's case, a few other medals as well), and good FDJ participation was the main part of your ticket in to the Communist Party.

Even then, actually joining the party was difficult. You had to submit an application, be supported by two party members, and then undergo a year of "candidacy" while your background (i.e. where FDJ membership was a big deal) was scrutinized. And not everybody was allowed in; Christianity was permitted in East Germany, but being a member of the church (or even worse, closely related to a pastor) was an almost guaranteed no. Academics had a slightly better chance, but still not particularly great.

The Stasi (and various Soviet equivalents) were very much a force at this time, though, and the Stasi in particular almost literally had ears everywhere. The Stasi was, essentially, a direct arm of the party itself, and if they got any evidence that you harbored any negative feelings at all towards the Party or country, you'd also almost certainly be denied entry (at the very least).

If you WERE allowed in, that opened all sorts of doors. If you chose to be a "career soldier (pledge at least ten years of military service, which would be another good thing to have on your record)," you were eligible to become an officer and/or "recruited" by the Stasi. If you weren't a career soldier but still a party member, you could attend university, or travel abroad, or pursue a political career.

As far as class divisions are concerned, I'll talk a little more about the NVA (army), which very clearly shows the distinctions. At the lowest level, you had people who dodged conscription (you could skip service on moral grounds), IF you wanted to spend the rest of your life in construction or a similarly undesirable job. Similarly, the least "politically reliable" soldiers ended up in military construction units.

Then you had your regular conscripts, who served for 18 months, then were considered "reservists" to be called up in case of war.

Then there were "volunteers" and "career soldiers," who pledged a longer period of service (five, ten, even twenty years) for increases in pay, and, for 10+ year "careers," permission to wear better quality, officer-type uniforms. That was your ticket towards a prominent and illustrious military or political career. This is where it's most defined, but class distinctions like this were kind of an extension of East German society as a whole. Many of these distinctions existed in the Soviet Union as well, but not always as clearly.

Follow up question: What was the situation like in then Czechoslovakia, one of the satellite states of the USSR? Was it the same, or were there any major differences?