As a general rule, a claim arising under a federal law can be filed in federal court without first filing in state court. This is called federal question jurisdiction.
But, state governments cannot, as a general rule, be sued in a federal court. You can such local governments in federal court and you can sue individual employees of state governments in federal court, but you cannot bring most kinds of claims, even if they arise under a federal law, against a state in federal court. This is basically a consequence of the 11th Amendment to the United States Constitution.
If you want to sue a state government for violating federal law, you must do so in state court. The only time a federal court can review that ruling is if the U.S. Supreme Court decides to review a decision by a state supreme court either on the merits, or by denying certiorari at the state level.
Also, in general, keep in mind that with some very narrow exceptions (e.g. patent and copyright cases), every case that can be brought under federal law in federal court may be brought in state court as well. The federal courts and state courts have concurrent jurisdiction over almost all federal claims (one or two kinds of federal claims can only be brought in state court, and federal court has exclusive jurisdiction over several kinds of claims).
If a claim arising under federal law is brought by a plaintiff in state court, and if the U.S. District Court would have jurisdiction over the claim if it were brought initially in the U.S. District Court (because, e.g., the defendant is not a state government), then a defendant in the state court case can remove the case to federal court. But, if a defendant brings a counterclaim arising under federal law in state court, the case cannot be removed to federal court.
For example, suppose that I sue you for failing to repay a loan under state law in state court, and you counterclaim against me for violating the Fair Debt Collection Practices Act under federal law. The case cannot be removed to federal court, even though it arises under federal law.
State court judges routinely interpret federal law. Most of the defenses in criminal cases arise under federal law. Many other substantive law issues arising under federal law are also considered in state court (e.g. the effect of a bankruptcy discharge in a state law debt collection action).
All federal judges start out as lawyers trained in both state and federal law and then practice law and become distinguished and not infrequently serve as state court judges before serving as federal judges. There are some subtle differences in procedure between state court and federal court, and there are certain kinds of cases that are only tried in federal court (e.g. immigration cases).