There are so many books about the island, and readers' tastes can be so different, that it's not easy to give definitive advice. I have a library full of non-fiction works on Skye, some good and others far from it. The ones I enjoy and return to again and again are:
 
 

For hillwalking, I think there's nothing to better Ralph Storer's "Skye - walking, scrambling and exploring"

Heavier reading, but quite fascinating is Alexander Nicolson's "History of Skye"

And if you want to know about the ghosts, witches, fairies, water horses and devils that inhabit this island, you'll enjoy the excellent "Guide to Mysterious Skye and Lochalsh" by Geoff Holder