Continuity mistake: In the scene where we see Buffy's mum drop her off at school, when Buffy is worrying that her dad will not show up, we firstly see Joyce's seatbelt is on. We then see her again and it is still on. However after the shot goes back from Buffy to herself, we can see her seatbelt is off. Then when the shot comes back to her again it is back on.
Revealing mistake: When Willow opens the door to the basement to look for Buffy, the shot looking up the basement's staircase shows Willow's shadow on the inside of the door. In the previous shot when she opened the door, it opens towards a wall and a staircase to the second floor. But there is no light source coming from the wall or the staircase, giving the impression that the basement was filmed on a separate set.
Continuity mistake: Buffy's history teacher is not the same person from episode 1 or 9, originally a woman, the teacher is now a man. Cordelia also notes Buffy doesn't know where the class is due to ditching, but we saw Buffy in her history class with Cordelia in The Puppet Show. How could Buffy not know where her class is only 1-2 weeks later?