What are the stages a case goes through after the supreme court agrees to hear it?

There are various steps that lead to a Supreme Court hearing. First of all, the case begins in the lower courts. If unhappy with the decision reached in this court, the case can be appealed before US Court of Appeals. If this doesn't go well, the defendant can request a rehearing or petition the Supreme Court. A petition of certiorari has to be filed, this will ask the Supreme Court to hear the case. At this point, the Court will either agree to hear the case, or reject it.