Does the Supreme Court have the power to review decisions originating in both the lower federal and the state courts?

Yes, under the right circumstances. The US Supreme Court can only review cases on appeal from, or when an appeal has been refused by, a state's "court of last resort" (usually called [state name] supreme court, but there are a few exceptions), if all state avenues of appeal have been exhausted and if the case involves a preserved federal or constitutional question. For an issue to be preserved, it must be raised at the trial level and all levels of appeal.

About two-thirds of the Supreme Court's cases are petitioned from the US Courts of Appeals Circuit Courts; however, under certain circumstances, cases may go directly from a US District Court to the US Supreme Court on expedited appeal. Expedited appeals most often occur if the case involves a high-ranking federal official, or challenges legislation in which Congress has specified an expedited appellate process.