No, it doesn't work quite like that. You learn how the laws work (the hierarchy of case law, levels of court, legislation, regulations etc) when to use which one, how to argue the cases, how to identify which laws apply etc. You also learn subject areas (contracts, torts, criminal law, trusts, corporations, corporate transactions, insurance, family etc) and to learn each subject you will study the applicable laws (legislation, cases etc), including the applicable Supreme Court cases. A really simple way of describing it is to look at the legislation and then find the highest level of court case that interprets or uses that legislation. There will be lots of cases on a particular area and then try to find the ones that are closest to the facts in your case. In doing this, you will be looking at Supreme Court cases, too. Some areas have lots of Supreme Court cases, like civil rights issues. So, in answer to your question, no, Supreme Court cases don't typically make up a class. They are interspersed throughout all your law school classes.
I hope that overview helps.