What kind of state court system is set up so that trial courts hears those cases arising within its own District, circuit, or country, no matter what the subject matter might be

The correct answer is: Trial court of general jurisdiction.

Explanation:

Every state has it's own state court system that establishes that trial courts with general jurisdiction can hear every criminal or civil case that is not designated to another specific court. Therefor trial courts are where most of the cases are filed and evidence is presented.

Trial courts with limited jurisdiction can hear a certain type of cases, for example family courts only hear cases involving families like custody, child support, adoptions, etc.