Here are some of the problems I have specifically with the first book.
The Heroine is helpless. (Now from what I make of the rest of the series, she becomes slightly less helpless once she becomes a Vampire--but I'm not 100% positive she does become one.) Bella, from whose point of view the books are written, is a klutzy teenage girl who does not have a very positive opinion of herself. This reinforces the less than appropriate egos of all of the women who read the book; girls tend to have notoriously poor opinions of themselves.
Edward existence is an affirmation of her self worth. Again, her own worth does not come from herself; it requires someone else to tell her that she's actually worth something. To make things worse, she doesn't believe him. How many of us have a friend with a horrible self image, who never can take a compliment when we try to convince her otherwise? It is books like Twilight we should blame.
As I mentioned originally, the overarching summary of the problems is the stereotypical gender roles. Any of these arguments relate to this subject. Now, I don't think there's anything wrong with these stereotypes myself; the problem is when characters are entirely the stereotypes, and Meyer never goes beyond the typical with her characters.
The other major issue I take is with the portrayal of this "predestined" love which our culture seems to have an obsession with. There is never a problem with the love, the only issue their relationship has is due to Edward's being a vampire and Bella's possible exsanguination at his hands. However, even that risk is portrayed unconvincingly. Real world love doesn't work like that, yet our modern culture continuously dictates otherwise. People are starting to believe this, and therefore at the first sign of trouble in a relationship, they jump ship. Want to know why the divorce rate has sky-rocketed?
Above and beyond this first complaint of the impossibility of their relationship, it is also an unhealthy kind. A certain phrase said by Edward jumped out at me and has remained ingrained in my head; in response to Bella's "I love you," he replies with "You are my life now." OK. "You are my life now" is NOT "I love you." In fact, it is unhealthy; it is dependency. A relationship in which both are reliant upon the other is impossible, a paradox. If anything goes wrong for one, it instantly spirals out of control into hell (and we know things never are continually good.) A relationship in which one is reliant upon the other is bad also: the dependent one will suck the other dry.
So the book tells us the relationships which we have which can work and which are good are not any of those, and instead insists a broken hopeless impossible alternative is right.
But if you're into not analyzing what you've read and wont ever notice those things, it's a mildly entertaining book.
Also Kevin, since I know you'll read this, look for me on AIM I need to talk to you (soon too please).