The way most cases reach the supreme court is

There are a few ways: 

1) The most common is on appeal from state courts. A case originating in state court must work its way through the state court system up to the state's court of last resort (i.e. state supreme court), and then it can be appealed to the U.S. Supreme Court, but only if there is a substantial question involving a question of U.S. constitutionality. 

2) On appeal through the Federal court system. A common route for a case involving Federal laws and the U.S. Constitution is for it to be first tried in the U.S. District Courts, and then appealed to the U.S. Circuit Courts of Appeals. The party losing at the Circuit Court may then appeal to the U.S. Supreme Court. 

In each of these two situations, the Supreme Court has the option to deny a hearing for the appeal. 

3) There are a limited scope of cases that can go directly to the U.S. Supreme Court without having to go through the lower court systems. This is not common at all, but is provided for in Article III, Section 2 of the Constitution

The most common way for a case to reach the Supreme Court is on appeal from a circuit court. A party seeking to appeal a decision of a circuit court can file a petition to the Supreme Court for a writ of certiorari.