I think Buffy is one of TV’s all time greatest main characters; on the one hand she is an ordinary high-school student, concerned with homework, boys and every other high-school trope you can think of. On the other, she is the teenage incarnation of Ellen Ripley (from the Alien films) and vampire slayer extraordinaire. I think it is this combination that makes her so cool. I’ve always liked female heroines (side-effect of playing Tomb Raider when I was young?), but what makes Buffy particularly special is the fact that she is very human-she is far from perfect, flawed, but as a result, highly relatable.
I think Buffy Seasons 1-3 are best for demonstrating her character, but Season 6 (despite being the least-good by a long shot) is also good showing her coming to terms with having adult life and problems and ostensibly failing….but that’s part of who she is; she might be the Slayer and saviour of the world several times over, but she’s still human.