In my opinion, the best, albeit somewhat dated, graduate-level solid-state textbook is Ashcroft & Mermin's Solid-State Physics. Kittel's Introduction to Solid-State Physics is more elementary and more up-to-date, but it is also less general (most analyses are in 1D, whereas Ashcroft & Mermin perform most analyses in 3D), and has less depth (some topics that Ashcroft & Mermin analyze quantitatively in a chapter are briefly described qualitatively in a page or less).
I personally would recommend first reading Kittel to learn the concepts, then reading Ashcroft & Mermin to bring your understanding of the concepts up to the graduate level. Both books will require careful reading in order to learn the material--I used to hate Kittel until I read it carefully for a solid-state course last semester, whereupon I came to appreciate how pedagogical his presentation is. Finally, be aware that the most recent (eighth) edition of Kittel has typos that are not in the previous (seventh) edition.
