Continuity mistake: When Buffy and Angel get to Buffy's house at the beginning, Buffy starts to use a first-aid kit to patch up Angel's wound. Angel's arm goes from down by his side to up on Buffy's shoulder between cuts. (It can be hard to see since it's close to the bottom of the frame, but it's noticeable. It seems like they cut out a few seconds to quicken the pace.) She's also a little bit closer to Angel between the same cut.
Continuity mistake: After Giles closes the library for "filing, " we see Buffy grab a crossbow off a shelf. It is upside down when she grabs it. Instantly between cuts, it is right side up. You can tell from the grip thingy that sticks out from it. (It looks like she starts to rotate it in the first shot, but it's instantaneously flipped over in the next).
Chosen answer: From what we see in the series, certain girls are identifiable as potential Slayers - Kendra clearly fell into this category. The identification method is presumably mystical in nature, but the Watchers' Council are pretty effective at that sort of thing, so they're quite good at tracking down the potentials ahead of time. Not perfect, though - it does appear that Buffy herself may have slipped through the net - certainly she had no inkling of what she was until she'd already taken on the role of Slayer. It is possible, however, that this was actually cultural - an American family would hardly be likely to turn over their daughter to some strange man for 'training', so the Watchers might have chosen to keep an eye on her covertly, whereas some other cultures (like Kendra's Jamaican parents) might be more willing to believe.
Tailkinker ★