Is there a website where you can read full books old and new online for free?

Project Gutenberg is a site that provides electronic versions of out of copyright books. Because it only deals with out of copyright books, you won't find new books there.

Some authors provide electronic versions of out of print books - for example, James Follett's website provides downloadable versions of some of his out of print books.

You should be able to find these two sites by searching for project Gutenberg or James Follett.