I don't think I can give you the full answer but I hope it'll be better than nothing.

One of the reasons for my inability to answer this fully is the fact that the Soviet Union was here for 70 years, and USSR in 1940-1950 and USSR in 1970-1980 are very different from each other. For example, there were free schools in the USSR, but only for the first 8 years, after that parents had to pay if they wanted their kid to continue to go to school for two more years (obviously, it was much easier to get in to the university after 10 years in school in comparison to 8 years and some other education). This was changed only in 1956.

So, my answer would be about the last 15-20 years of the Soviet Union (1970-1980). If someone wanted to get into the university, she had to finish her school and pass the university tests, that's it. If your score was better than minimum score needed for this university this year, you're in (obviously, minimum score was dynamic, and it's value was different every year, it depended on the number of students that university could take in and the number of students who wanted to learn there).

Although, first 10 years of school were already free at this time, a lot of kids just dropped out after the first 8 years, because they didn't want to be students anymore and/or because they needed to work to help their families. Also, there was a professional education (sort of schools, but in addition to school curriculum you'd get some working profession there; it seems like a good idea in theory, but in practice the quality of education was usually worse than in real schools).

High education (universities, institutes and so on) was free. More than that, students, especially ones with good scores, would get some monthly stipend from the state. Students from other cities could get a place in the dorm (also, free). But, as you can see, there already was some inequality. It was much simpler to get in if your family wanted you to learn and your family could afford it which wasn't always the case.

Some categories of students could get in without test exams, or with somehow simplified procedure. For example, it was much easier to get into some Moscow university for some wanna-be-student from so-called national republics (regions that were formed and called on ethnic base) than for someone from Voronezh (Russian city). Also, there were some simplified procedures for people who good at sports, or for people who served in the army and so on.

At the same time, there were some restrictions. We basically don't have any real documents for that, these instructions were not put to paper as laws and such, but it's a known fact that it was quite difficult for jewish persons to get into some universities. Sometimes, it was just antisemitism, sometimes there was some rationale behind this politics (jewish persons couldn't be trusted because all of them want to immigrate to Israel, so we don't want them to learn some really important stuff). I'm not saying this rationale was good, I'm just trying to explain the logic behind such decisions.

So, yeah, the Soviet Union _tried_ to make the high education more or less accessible for all, but not always it was really successful. Actually, you had to be quite determined and resourceful if you wanted to get in the best universities: test exams were hard, and your school education often was not enough (so, your family had to prepare you for this, and it needed money for this and so on). Also, try not to be a jewish person, if you want to be a mathematician or physicist.

One detail, I didn't mention, is what happened after the graduation. Higher education in the USSR was free, but with a twist. After the graduation, student had to work in a place that was chosen for her by the state. And this place could be anywhere in the country. So, it was quite usual for some Moscow student to go to, I don't know, Siberia and work there for two or three years. The positive side of this: there wasn't an unemployment problem, you had a job right after the graduation. The negative side: you couldn't what the job was and where it was.