In the United States there are two parallel court systems. There is a federal court system, established by the United States Constitution, which is restricted to hearing cases involving federal law (issues arising under the federal constitution, laws passed by Congress, federal regulations, treaties) and some cases involving disputes between people living in different states. And then each state has its own court system, which can hear all cases except those which lie exclusively within the power of the federal courts.
The Ohio Court of Common Pleas and the Ohio Supreme Court are both courts in the Ohio state court system.
The "federal court in Columbus" is the United States District Court for the Southern District of Ohio, Eastern Division. It is part of the federal court system and it hears federal cases arising in the southern part of Ohio.
There are many more courts than most non-lawyers could ever imagine. In Ohio, we have municipal court, common pleas courts, courts of appeal (intermediate courts), the Ohio Supreme Court. Federally, you have the Northern District of Ohio, Southern District of Ohio, Sixth Circuit Court of Appeals in Cincinnati (covering federal appeals in Michigan, Ohio, Kentucky, and Tennessee), and of court the U.S. Supreme Court. There is also other "Article I" federal courts such as tax court, bankruptcy court, immigration court, and more state and federal administrative offices than I care to list here.
Federal court, which interprets federal laws and the U.S. Constitution, refers to U.S. District Court and the appellate levels, the U.S. Sixth Circuit Court of Appeals which is in Cincinnati, and ultimately, the U.S. Supreme Court in Washington, D.C. All parts of the U.S. are under the jurisdiction of some federal district court. In Columbus, that would be the U.S. District Court for the Southern District of Ohio, Eastern Division, which is located at 85 Marconi Blvd.
State courts have their own hierarchy of courts. There is a common pleas court in every county, and also courts that handle less serious matters like traffic offenses, which are called variously, municipal courts, mayor's courts, and county courts. Appeals from all of these courts are heard by the state district courts of appeal. Ohio has 12 appellate districts, and Columbus is located in a single-county district, the Tenth District Court of Appeals. Appeals from a district court of appeals are heard by the Supreme Court of Ohio, the highest court on questions of Ohio law.
Now, that's an explanatory primer of the court system, but I think what you really want to know is whether state or federal court is where you should go. Since all states are under the jurisdiction of the U.S. Constitution and federal law, it may be possible to enforce your federal rights in either state or federal court. For that, you should schedule a consultation with a lawyer.