American FactFinder - The US Census Bureau's portal for researching demographic, business, and other information. Useful for learning abouth the people and industries that may appear before the court.
The American Bench, ch. IV (right after the Alphabetical Names Index) shows Judicial Boundary Maps for the federal and state court systems. Federal district and state court boundaries are shown to county level. KF8700.A19 A54
Federal Judicial Caseload Statistics (from Administrative Office of the U.S. Courts) - open current report or archived reports to see list of detailed statistical tables showing caseload and other information broken down by district or circuit.
Lexis Context - start typing in the court name (ninth circuit) and select yours; you'll see an overview with numbers of opinions, by area of law for the top 10 (broadly defined) topics, number of opinions per year; on the Documents tab, you can filter down by date and topics to see, for example, how many opinions in the past 5 years related to deportation/removal (a subtopic of immigration law)
Annual statistics reports from the Judicial Council of California - 1998-present from California Courts website - (California) Supreme, appeals, and superior court trends, broken down by county, district and appellate division.
Additional Research and Statistics from the Judicial Council and Administrative Office of the Courts (California)
Lexis Context - start typing in the court name (vermont supreme) and select yours; you'll see an overview with numbers of opinions, by area of law for the top 10 (broadly defined) topics, number of opinions per year; on the Documents tab, you can filter down by date and topics to see, for example, how many opinions in the past 5 years related to estates and trusts
NCSC's Court Structure by State map - click your state for an overview of the appellate structure, click on each of the courts in the appellate diagram to see some details about the jurisdiction; in the court detail, you can also click through to the court’s official website and look for (usually high-level) statistical reporting, usually found under something like "Court Administration " or "Administrative Office of the Courts" > Reports/Publications > Statistics or annual reporting (e.g., New Mexico has this Statistics Addendum 2020)
Even if you are not interviewing for a trial court position, you can learn about the caseloads and demographics (including aggregate gender/racial/ethnic makeup of judiciary and other court officials) of the court system that feeds to the state appellate/supreme court level via the Court Statistics Project's State Court Organization Data interface (selected state coverage). It's expected that appellate court data will be added soon.