It starts with a petition. After a trial through the district courts to distinguish the facts, then a second trial through the appellate courts in order to determine whether due process was followed, a case can be petitioned to be tried by the supreme court. This is not guaranteed though. The Supreme Court only takes about 100-150 cases per year out of around 7,000 (www.uscourts.gov). Because the Supreme Court has "certiorari" or the power of the court to pick and choose which cases they take and do not take, each of the nine justices has a team of lawyers to filter through the cases and aid in picking which cases they actually try.
There have been just over 100 appointed justices since the establishment of the Supreme Court in 1789, each serving an average of 16 years. The SCOTUS is not always truly right and just though. In the Case of the Dred Scott decision of 1857 in Dred Scott v. Sanford, the Court ultimately decided that Dred Scott, a slave at the time, was not entitled to his freedom even though he resided in a free state and stated that the United States Constitution did not include people of African descent in the term "American Citizenship". Fortunately, SCOTUS has since redeemed itself with cases such as Obergefell v. Hodges, Loving v. Virginia, and Griswold v. Connecticut. Throughout history, the Supreme Court has made it's mark on history and continues to do what's best for the citizens of the United States.