Did the case of Plessy versus Ferguson go straight to the supreme court after it went through the criminal district court for the parish of Orleans?

No, the case of Plessy v. Ferguson did not go straight to the Supreme Court. It first went through the Criminal District Court for the Parish of Orleans, where Homer Plessy was convicted for violating Louisiana's segregation laws. After that, the case was appealed to the Louisiana Supreme Court before ultimately being taken up by the U.S. Supreme Court, which issued its decision in 1896.