I agree with what some others have said against showing "Zuko Alone." While it's definitely one of the best episodes, you need to have seen more of the show to understand and appreciate Zuko's character arc. Here's what I would recommend:
Season 1: "The Storm"
Season 2: "The Avatar State"
Season 3: "The Avatar and The Firelord"
I chose these because I think they're the easiest for someone new to the story to understand. But if you want to show your boyfriend something with a bit more action, then I'd recommend "The Waterbending Master" from Season 1(which should hopefully get him interested enough to watch the two season finale episodes that follow, like you mentioned). Also "The Blind Bandit" from Season 2, as others have recommended(along with "The Chase.").
It's great to hear that you're trying to get your boyfriend hooked on Avatar.