I. Buffy feels responsible for Jesse and is mad at herself for not saving everyone. Xander and Willow don’t hesitate to help in any way they can but Buffy doesn’t want to put them in danger. She was most likely trained by her first Watcher that the slayer does things alone (an idea she’ll later reject).
II. We don’t see who turned Jesse, but we can assume it wasn’t Darla because of how seriously she takes that process (from Angel).
III. Let’s talk about Angel for a moment. From a writing standpoint I can see they were trying to make him mysterious and conceal his backstory/vamp status but from an in-story point of view he’s a jerk. He meets Buffy at the entrance to the tunnels but doesn’t go down with her. Angel spent centuries with Darla and knows the Master well, so he’s aware of exactly how dangerous this situation is. Yet he chooses to stay behind. Why? The only thing that makes sense is that he doesn’t want Buffy to know about him and his past. He’s got a crush on her and is choosing to conceal the not so great bits about himself until he’s reeled her in. He’s more concerned about getting what he wants from her/molding her perception to his liking than her safety. Eh.
IV. Cordelia is wasting no time in making Buffy the pariah of the school. Willow gets very defensive and causes her to delete the project she’s been working on. Some cite this as “early evidence of Willow’s bullying/controlling behavior” but I think that’s going a little far. Willow has endured countless torment from this girl and now she’s attacking the person trying to save Jesse. It’s understandable.
V. The Master came to Sunnydale looking to open the portal and bring the old ones back that originally ruled the earth (the “pure” vampires). An earthquake somehow trapped him like a cork in a bottle. This is an early sign of his dedication to the pureness of vampirism. He’s always in vamp face and his mouth looks stained with blood.
VI. When Joyce finds out Buffy skipped class she decides she doesn’t want her going out again. She says “the tapes all say I should get used to saying (no)” which was amusing. Buffy ignores her and climbs out the window. She didn’t have much choice there but she did seem to prefer having her mother’s permission.
VII. Vampire Jesse still desires Cordelia which is an early hint that becoming a vampire doesn’t completely change you. Giles contradicts this when he tells Xander that when he looks at Jesse he’s not looking at his friend, he’s looking at the thing that killed him. Xander doesn’t want to believe this and tries to see a sign of the old Jesse but ends up staking him. I think this first experience with vampires was extremely important for Xander’s character. From here on out he almost has to believe vampires are irredeemable creatures of evil; otherwise he killed his best friend. There isn’t much focus on how Xander dealt with losing Jesse but he clearly grew to resent what took him away.
VIII. Buffy uses her wit to kill Luke by pretending she’s going to burn him with sunlight. Still the same Buffy. Well, not quite. She’s super girly in the first couple episodes with her mini skirts and lollipops.