There are two other categories of books that might be useful to learners. These books that are familiar to and enjoyed by adolescents that are not considered great literature or assigned as school reading.

a) Books that are read to children by adults, and

b) "Young adult" fiction.

These are not necessarily easy English but they are not terribly difficult, either. I will mention a few at random. Although they are read to children, or read by adolescents, they are not "childish" at all. 

1) The Twenty-One Balloons, by William Pène du Bois.

2) The Bad Beginning, and a dozen other titles in a series, A Series of Unfortunate Events, by Lemony Snicket.

3) Half Magic, by Edward Eager, and other books by the same author.

4) The Time Machine, by H. G. Wells.

5) Charlie and the Chocolate Factory, by Roald Dahl, and other books by the same author. 

6) Charlotte's Web, by E. B. White.

7) The Harry Potter series, by J. K. Rowling.

8) The trilogy--or, rather, three-volume novel--The Lord of the Rings, by J. R. R. Tolkien.

9) The Lion, the Witch, and the Wardrobe, and the others in the Chronicles of Narnia series, by Tolkien's friend and colleague, C. S. Lewis

10) Have Space Suit, Will Travel by Robert Heinlein

Some books I read or remember others reading

The Scarlet Letter

Lord of the Flies

Othello

Romeo and Juliet

Macbeth

Fahrenheit 451

To Kill a Mockingbird

Adventures of Huckleberry Finn

Pride and Prejudice

Animal Farm

1984

Brave New World

In my opinion, the Bible and Shakespeare are not good choices. Yes, we read them in high school. (In my public school we read "the Bible as English literature.") They are difficult for native speakers. Shakespeare, and the culturally important "King James" version of the Bible, were written in about 1600 and the language has changed. Bookstores now offer books that are called translations of Shakespeare. Many churches now use "modern translations" of the Bible. Both use certain verb forms (familiar 2nd person) that are not used today: "Thou art a scholar" instead of "you are a scholar." 

Here are some books that meet these criteria: 1) they are often assigned reading in high school, 2) they are in intermediate-level English, and 3) I love them:

Black Beauty: The Autobiography of a Horse, by Anna Sewell.

To Kill a Mockingbird, by Harper Lee. 

Animal Farm, by George Orwell. But you will need to read some explanations to understand it. It is an allegory of the Russian Revolution. It expresses George Orwell's political opinions. He thought the rulers of the Soviet Union had betrayed the ideals of the Revolution. It is a readable and entertaining story without understanding the history, but I think the history is important. It isn't just a vague allusion, the characters and events match up, and Orwell even wrote to his publisher saying something like "I had to rewrite the part about the Battle of the Windmill because I can't find evidence that Trotsky was really at thus-and-such place at thus-and-such time."

Fahrenheit 451 by Ray Bradbury.

The Bible and Shakespeare are terrible recommendations! Read what you enjoy.

I would suggest watching English language films with English captions. That gives you listening and reading practice. "Dumb and Dumber" is going to sound a lot more like current English than those antiquated books.

Another thing to keep in mind when choosing books is that English literary language changed around the 1930s. It became simpler, more direct, more conversational, less colloquial. Always check when a book was published. The language of most novels written before 1900 is going to be challenging. Writers wrote long, complicated sentences. 

Because of copyright restrictions, people like to recommend books written before 1923 because they are out of copyright and thus legally available without cost. 

The best books for English learners, in my opinion, are more modern and therefore (unfortunately) hard to find legally without cost. Do not start out with Charles Dickens and Jane Austen and Emily Bronte and George Eliot!