While I'm no lawyer, I believe you may petition the U.S. Supreme Court to hear your case only after exhausting all other avenues of appeals (local, state and federal appelate courts).

It is usually taken to the US Supreme Court because it has been requested to be there. Cases have to be taken thru the chain of courts to reach the US Supreme Court. The only exception to this is if the Government is prosecuting a federal criminal. Most often, if a case has no federal elements, whatsoever, it will never see the inside of the Supreme Court. But if it has SOME federal merit, then it will go there if it can't be resolved in lower courts.

A case can get to the Supreme Court through [1] an appeal of a federal question from a case in the highest State court, [2] an appeal from a lower federal appellate court. In very limited situations a case can be brought directly to the Supreme Court (e.g. disputes between two states - that is, between the states themselves). If the case is from a state supreme court, then there must be a federal legal question in the case which if reversed would alter the result of the case. The Supreme Court is almost never obligated to grant an appeal, unlike most appellate courts. In fact, the Court hears only a small fraction of the cases that petition for cert. One of the major reasons that the Supreme Court hears a case is to resolve a split in opinion among the federal circuit appeals courts. For example, when the 5th Circuit Court of Appeals (covering the states of Texas, Louisiana, & Arkansas) decides an issue differently than the 2nd Circuit Court of Appeals (covering the states of New York, Connecticut, & Vermont) has already decided, then the Supreme Court will often hear the appeal in order to settle the division, ensuring consistency in application of the law.

You must file a writ of certiori -- and the Supreme Court grants less than 2% of those filed each year.