Your appeal from the NM administrative agency decision generally goes to NM state district court. In this role the district court is functioning like an appeals court, not a trial court. As a result, there would be no jury and no trial. Rather, the proceeding is done as a review of the record of the administrative proceeding. You don't get to introduce any new evidence (and neither does the agency) in the district court and the decision on the matter is made by the judge, not a jury.
Then the losing party in district court may appeal the matter to the NM Court of Appeals, if he or she has grounds for appeal. But, as indicated in what you quoted from the court web site, this appeal is discretionary. This means that while the losing party in the district court may file for appeal to the Court of Appeals, the Court of Appeals does not have to hear the appeal. It can say it refuses to hear the case, in which case the district court's decision would remain.