Most states have a one-tier system and the decision of the hearing officer is the final agency decision. (If the state has a two-tiered system, the hearing officer's decision can be appealed to the second tier hearing officer or state review officer.)
The hearing officer decision can be appealed to court. The appeal can be to state court, but it usually is to federal court. The first level court in the federal system is the U S District Court. This court can accept additional evidence, but usually they consider the administrative record compiled before the hearing officer. (The due process hearing is very important.) The court will usually accept the hearing officer's findings of fact unless there is a pretty good reason not to do so. The judge will, however, apply his own interpretation of the law to the facts.
The decision of the District Court can be appealed to the federal circuit courts of appeal. Each state is in a federal circuit. I am in the Fourth Circuit. New York is in the Second Circuit. California is in the Ninth Circuit. The circuit courts usually only consider legal and not factual issues.
A party can then ask the U S Supreme Court to accept a decision by a circuit court of appeal. The high court does not have to accept such cases, but may do so if the Justices decide to do so.
Decisions by the Supreme Court and the circuit court for your area and your U S District court as well as any decisions by your state high court or appellate court are binding precedent. Decisions by hearing officers or courts not covering your area may be used a helpful and persuasive reasoning, but they are not binding.
Although legal citations may look confusing they are useful. Even if you don't know how to look up a citation, if you type the citation or the names of the parties into a good search engine, you can often find news articles or other websites that provide more information.
Welcome to the world of legal research! Although I still believe that everybody should have their own attorney, I hope that this crash course was helpful.