New Moon starts where Twilight left off, which is to say the same place every one of these movies starts and ends because no matter what changes, no one really grows or learns anything because of it. It's Bella's birthday and she's freaked out about being a year older than Edward, even though she isn't really, because he's 109. No one actually seems to think about emotional age, except for Edward who's so emotionally stunted that he's afraid Bella will outgrow him someday. He's got nothing to worry about, because Bella doesn't care to think of silly little things like personal growth.
Anyway she celebrates her 18th with Edward's vampire family, and his brother tries to eat her, so Edward breaks up with her and leaves her alone in the woods. Happy birthday, Bella! She sits in front of her window watching the world go by for a while, and then eventually she decides to go and try to get kidnapped and locked in some dude's basement because she discovers that she hallucinates a vision of Edward when she does dangerous shit. I think that's one of the stages of grief. Anywho, she buys two broken motorcycles and has her friend Jacob fix them so she can hallucinate some more. Jacob's destined to turn into a werewolf and have his furry little heart stomped on by Bella, which is kind of a shame, because unlike Edward, Jacob actually does stuff with her, like share a hobby and talk about something besides how desperately he wants to kill her and feast upon the remains.
Then Bella goes cliff diving and Edward's psychic sister has a premonition that she threw herself off a cliff for suicidal purposes and not just to recreationally make herself hallucinate. And now Edward thinks Bella killed herself and is off to kill his own self by breaking laws and making the vampire police kill him. Being a vampire sounds pretty awful, especially for Edward because he's basically doomed to be a high school student forever and he can't eat people and now he can't even kill himself without going all the way to Italy to jump through hoops with the vampire police.
But Bella gets there in time and saves him and there's a showdown with the vampire police and they say that Bella has to become a vampire. Edward's brother later says that's a great idea so he can stop wanting to kill her so damn much. Then Jacob flips his shit and Bella says she loves him, but not as much as she loves Edward because she's kind of a terrible person.
New Moon is better than Twilight, but only by a little. There isn't much romance, and Bella spends a lot of it being horribly depressed. However, the werewolves are fun to watch and I've always liked Jacob's character, and it's nice to see Bella pal around with someone who doesn't want to kill her. It has a little more plot than the first movie, but the reason why the first two movies seem to be plotless is because almost all the plot is Bella's internal conflicts. The things that happen outside of Bella's head really do pick up steam from here on out, and even the book counterparts of Eclipse and Breaking Dawn are more exciting.