In my public schooling, I had two year-long classes about the structure and function of the United States government. One was in middle school and one was in high school; I was not required to take any course on the subject as part of undergrad. Neither course taught me much more about courts than that the Supreme Court is the highest court of appeal and interprets the Constitutionality of laws.
Courts are complicated and laws and regulations are different from jurisdiction to jurisdiction, but there are some fundamental, low-level principles that apply to proceedings across most (possibly all?) courts in the US. An educated citizenry in the US would, ideally, have at least some exposure to these fundamentals; these basic principles (which would include at least a little on contracts) are too important for the vast majority of the population (i.e. almost all non-lawyers) to not understand.
I told a lawyer friend about this idea and she strongly agreed. She said that her sister had taken an undergrad class which roughly fit my description-- "pre-pre-law" was how she described it. Her sister's evaluation of the course was that it's description of legal principles and their operation in courts helped explain a lot of the mechanics of how government, private institutions, and society function.