Once a Custody order has been determined in your county court, can you appeal to a higher court in California?

Generally-yes.
But it is not so easy task.

It will require ( or should) an attorney, you can not appeal just because you lose, First you have a limited amount of time to appeal, and then there has to be grounds for it, something that shows a reason for error of the first court.

Appeals of any kind must be filed in a timely fashion. And the appeal must show the judicial error on the part of the judge. Just because you lost or don't like the outcome is not a good enough reason for an appeal. And lots of on point case files must be shown to be in accordance with your appeal as well. An attorney needs to hendle this and an appeal will cost you probably in the vicinity of several thousands of dollars for starters.