A similarly between the federal and state court systems is that both have

Federal and State courts both have their own court systems. In Each court they usually have Plea Bargains and Bringing charges against you.

Both state and federal courts of appeal are appellate courts. That is the main similarity between them. State courts of appeals hear appeals from decisions from trial courts in that state, while federal circuit courts of appeals hear appeals from district courts, the trial courts of the federal system.