Revealing mistake: In "To Shanshu in LA" in season one, as Angel is pulling out of parking lot to follow the W&H Trucks, you can see his reflection in his rear view mirror (vampires have no reflections). Also odd that throughout the first season the rear view mirror is removed in some episodes. (00:31:19)
Factual error: According to the show, Angel learned quite a few languages when he was evil. However, when he tries to start a conversation with a red-faced demon, he makes terrible mistakes. He said "Hablo Espanol?" which literally means "Do I speak Spanish?" and he said something like "Are you from Byelorussia?" in a tongue that hardly resembled Russian, although according to the captions, it was said in Russian.
Continuity mistake: In Season One Finale, "To Shanshu in LA" when Wes is lecturing Cordy on wants and desires, the number of bites taken from the doughnut changes, in one instance it breaks in half, most visibly when Wes takes a bite from Cordy's doughnut on the angle change. (00:13:31)