About 'Welcome to the Hellmouth' :
The series premiere begins at Sunnydale High School, where a boy breaks into the school during the night with a seemingly reluctant girl, promising her mischief and therefore fun. Nervous and on edge, the girl thinks she heard something and fears something is there. The boy calls out but gets no response, during which the girl's face morphs into that of a vampire, and bites the boy. The girl, we will later find out, is Darla.
Buffy Summers has a nightmare the morning of her first day of school. Her mother drives her to the school and encourages her to think positive. Inside the building, Principal Flutie tells her she will start with a clean slate. He reconsiders that after realizing that Buffy burned down her previous school's gym because "it was full of vampi—…asbestos."Buffy exits the office and bumps into another student, spilling the contents of her handbag over the floor. Xander sees that and helps Buffy, introducing himself. She leaves without her stake, which Xander pockets because he called out to her, but she had already walked away. In history class, Buffy is helped by Cordelia, who afterwards tests her "coolness factor," skipping the written as Buffy had just moved to Sunnydale, California from Los Angeles. To Buffy's horror, Cordelia humiliates an awkward Willow at the water fountain. Inside the library, Mr. Giles places a book titled Vampyr in front of Buffy after realizing who she was. A stunned Buffy makes a hasty exit.
Buffy, Willow, Jesse McNally and Xander meet during a break, and Xander returns the stake. Buffy claims it is standard self-defense in Los Angeles. Cordelia appears and tells Buffy that gym is canceled "due to the extreme dead guy" in Aura's locker. Buffy asks whether there were marks on the body, freaking out Cordelia. Buffy forces her way into the locker room, examines the body, and finds the characteristic puncture wounds of a vampire on the neck.
Buffy returns to the library and confronts Giles, who informs her that he is her Watcher. Buffy refuses to accept her calling as a Slayer, since it had gotten her kicked out of her previous school and cost her her social life. After they leave the library, Xander emerges from behind the shelves, having overheard the strange conversation.
That night, en route to her first visit to The Bronze, the cool hangout in Sunnydale, Buffy meets a mysterious, handsome stranger, who warns her that she is living on a Hellmouth that is about to open, and that "the Harvest" is coming. He also gives her a large silver cross. It isn't revealed until the next episode that the stranger's name is Angel.
In The Bronze, Buffy meets Willow and encourages her to seize the moment because tomorrow she might be dead. She finds Giles and tells him about Angel. Giles tells her to learn to hone her skills to sense vampires anywhere. Buffy uses her fashion sense to pick out a vampire in the club and is alarmed to see Willow leave with him. She loses them and is surprised by Cordelia, nearly staking her. Cordelia immediately calls her friends to tell them about it. While Buffy looks for Willow, Jesse chats up Darla at The Bronze. Buffy is stopped by Xander, whom she convinces to help search for Willow.
Meanwhile, under the streets of Sunnydale, The Master is woken by lesser vampires from a long sleep to prepare for the Harvest. He sends Luke to fetch young blood.
Willow's new acquaintance takes her to a crypt in a cemetery, where they are joined by Darla and Jesse, whom she has bitten. Buffy and Xander arrive. Buffy kills Willow's vampire. Xander and Willow help Jesse, who has been weakened, flee. Luke takes Darla's place in the fight so she can help catch the kids. Luke throws Buffy in a stone coffin and is about to move in for the kill.
Info from the Buffy Wiki.