The Elegant Universe has a special place in my heart because I read it the summer I fell hard for physics and decided to change majors. That said, it’s primarily an account of string theory and a quite biased one at that. I’d suggest Fabric of the Cosmos first since it’s more general, less controversial, and frankly just more enjoyable. If you do read The Elegant Universe though, be sure to pick up something more critical to balance Greene’s gushing love poems for string theory (like Lee Smolin’s ‘The Trouble with Physics‘). I read both Smolin’s book and The Elegant Universe over a year and a half ago, so if you want my immediate thoughts after reading them, check out the links!