Economics classes in communist countries were generally focused much more on microeconomics and industrial planning, while U.S. courses focused on macroeconomic theory and market forces.

For example, compare U.S. based microeconomics courses(both graduate and undergraduate levels) to Eastern European microeconomic courses. In the U.S., the general focus is on consumer behavior at a granular level. We apply quantitative methods to individual behavior (e.g. game theory) to understand how individuals behave in a market. We then use these theories (e.g. rational consumer) as a foundation to scale up to market behavior analysis.

However, in many eastern European countries, the focus of their curriculum was much more "business" oriented. Microeconomics courses in countries such as Russia focused on industrial planning. They used microeconomic tools such as marginal cost and revenue and so forth to determine the most efficient way to produce and price goods. This ideology is very quantitative and we still see artifacts of this skill be necessary in Eastern European and Russian mathematics curriculums (which would be a whole other topic).

So to keep things short and general, most western economics curriculums focused on market forces, consumer behavior, and demand while communist countries generally focused on industrial planning, microeconomics, and labor economics.

Here is a source telling you a little bit more about the above: https://www.gwu.edu/~ieresgwu/assets/docs/demokratizatsiya%20archive/03-4_BrueMacPhee.PDF

On an additional point that I can't find a cite on, I had a macroeconomics professor in college named Richard French from George Washington University who told me that he was one of the first economists to be invited over to Russia to teach and help develop a macroeconomics curriculum. He told me that the majority of the courses were much more based on industrial planning and that the universities/governments really didn't concern themselves with the study of market principles and behavior. Again, I don't have a source on that, but I had a lot of conversations with him about the subject so if you'd like to hear more just ask.

In a rather odd coincidence, I happen to have a copy of a 1920s Soviet political economy textbook which had a syllabus in the pages! Hooray for primary sources!

Here we go!

"So, what would an economics student in a communist country, say in the Soviet Union, learn and what would be something that we'd learn in common and somethings that would be different?"

They learn economy as if it is all only about Marx's theory and as if Capitalism does not exist at all. You learn about classes: bourgeoisie and proletariat. You'd learn economy in connection to history and philosophy, you learn this basic concept of "the plan" and planning.

They rarely see how things are in capitalists countries, more concentration on "the right way" of command administrative system. Actually even now in Russia these views are not entirely gone and many old econ teachers still bring up some soviet things and mix them up with classic capitalism. For example the theory about quality (it's called there - quality management), production theories (pure math in Russia) and other areas have a connection to old USSR economy views. I was lucky to have a chance to compare because i finished University in Russia (econ major) and then i graduated from an american college (finance major) and even now i see a huge difference. Russia is very weak on economy, extremely weak. And it is all going from the way they teach it: super boring, they don't know how it works and how it supposed to work (capitalism), there is no practice at all. This is not true for huge and modern Universities in Moscow but in a province econ education is very week this days, even in modern Russia. Things were much worst in soviet union. No one in USSR in the right mind would choose to study economy, economy was strongly connected to politics. In USSR they studied practical things and concentrated on that, that is why soviet legacy could produce airspace ships and good weapons. But when it comes to "how to make money" or "how economy works", USSR had no clue.