Don't know about others, Atalanta, but I use the Penguin Classics version by Tarif Khalidi. It has a very good introduction. Note that today's version is not constructed in chronological order, so it doesn't make sense as a narrative. Ideally you should have a commentarty. Haven't got round to getting one myself - local Islamic book store didn't have one last time I was in there!
I recommend you read a biography of the Prophet, too. Karen Armstrong's Mohammed is a a fulsome hagiography, but is well-written and gives a good sense of what Islam is, certainly not "just another" religion, but a political project.
The main division in Isla - Shia versus Sunni - is mainly political, too - an argument over succession following Mohammed's death rather than disagreement over points of doctrine.