Many times, my daughter will pick out easy books at the library to read. I just let her pick out one or two books, whatever she wants, as long as it doesn't have any objectionable content (I don't tolerate rudeness, which seems to be in everything, or things that go against our beliefs). I bring home chapter books for her and put them in the book basket. I don't demand she reads them. However, she loves to read everything. She'll excitedly pick them up and read most anything there, even non-fiction books for "school." I keep out a couple of the higher level books for read-alouds. If I put those in the baskets, she'll read them and then not want to read them with me!
It is getting so difficult to find books that challenge her with appropriate content *sigh* I use the list mentioned in the post above (1000 good books) to try to pick out things. At her age, almost 7, I want to protect her innocence as long as possible. She also has lots of nightmares, like her mother, so I don't want to select books that fuel the problem.
On a side note, I was also worried about reading Charlotte's Web with her. I wanted to take her to a play production put on by children about it, so we went ahead and read the book. Amazingly, it didn't seem to bother her at all. Right now, I'm avoiding Black Beauty because it seems a little too graphic :p.