I've read all four books plus the extras at the website. Here are some of my disjointed thoughts.
Overall I enjoyed the story. I had a hard time putting it down.
I liked seeing the strength of the family, trust, loyalty, love and humor. And these qualities were even able to affect the feelings between natural enemies. (I'm trying to keep this general in case you haven't read the books yet.)
Things I cringed at...but it offered nice talking points with my girls:
works righteousness (but I wasn't reading this story for theological purposes),
some intimate bedroom scenes,
and a seemingly unhealthy "devotion" to a boyfriend.
Don't miss the extras for New Moon. Meyer has three snipets of the story from three other points of view. Jacob is a "crack-up". [NB: pgs 250-251 of Breaking Dawn]
I really liked reading the story from Edward's perspective. He had a lot more thoughts going on in his head than Bella who seemed kind of hung up on age and beauty. Hope Meyer decides to finish that book.
Is this saga a "classic"? My unprofessional opinion is "no".
Did I enjoy reading it? Yes.
Some are trying to compare this saga to Harry Potter. I'd say there really isn't any comparison. The Harry Potter novels are much deeper with more layers of things to experience on each reading. More mystery and surprise twists too.
If I had to apply movie ratings to these series of books....
I would rate Harry Potter as PG and Twilight as PG-13 (because of the bedroom scenes).