Assuming a case is capable of being heard by the U.S. Supreme Court, the first step, most of the time, is to file a lawsuit in your local state or federal court. The trial judge would hear evidence and consider legal arguments from each side before making a decision. If the judge decides all or part of the case against you, you can then appeal the case to a higher court. When you have appealed as far as possible, you can consider appealing to the U.S. Supreme Court. -
The beginning process is to file a notice of appeal, rarely does a losing party have an automatic right of appeal. Usually there must be a legal basis for the appeal, an alleged material error in a trial, not just the fact that the losing party did not like the verdict.
This filing marks the beginning of a time period in which the appellant must file a brief, a written argument containing their side's view of the facts and the legal arguments upon which they rely in seeking a reversal of the lower trial court. The appellee then has a specified time to file an answering brief. The appellant may then file a second brief answering the appellee's brief. A United States appellant court sometimes will make their decision only on the basis of a written briefs.
In a civil case, either party may appeal to a higher court. In a criminal case, only the defendant has a right to an appeal in most states. (Some states give the prosecution a limited right to appeal to determine certain points of law. These appeals usually occur before the actual trial begins. Appeals by the prosecution after a verdict are not normally allowed because of the prohibition in the U. S. Constitution against double jeopardy, or being tried twice for the same crime.)
Criminal defendants convicted in state courts have a further safeguard. After using all of their rights of appeal on the state level, they may file a writ of habeas corpus