It's a beautiful book, I was a little late to the party, read it and simultaneously listened to the audiobook by Armie Hammer, having watched the movie before.
The book is from Elio's perspective and Armie who plays Oliver, in his melodious voice reads it, which is bit conflicting at first but then adds whole other dimension to the book.
So I'll recommend doing it in this order:
Watch the movie, listen to the audiobook while reading the book, because Armie's voice is just a cherry on top of this amazing and heart touching piece of literature.