Topoi begins with an introduction to category theory and a steady build up to explaining how sets — or a generalisation thereof, what is known as a topos — can be defined without the concept of membership. This is well-motivated, with Goldblatt using analogies to set-theoretical ideas which might have been disdained by a category theory purist.
Goldblatt proceeds with more or less independent chapters taking a categorial approach to different facets of mathematical logic: classical logic and semantics, the algebra of subobjects (implication and extensionality), intuitionist logic (Heyting algebras, Kripke semantics, and a five page history of Intuitionism), functors, set concepts and validity, elementary truth, the category of sets, and arithmetic.
The last third covers local truth (Grothendieck topoi, geometric modality, Kripke-Joyal semantics), adjunctions and quantifiers, and logical geometry. Some of this is considerably more difficult — I confess to skipping parts of it — but it remains well-motivated and Goldblatt is willing "to take an approach that will be more descriptive than rigorous".
Originally published in 1984 but understandably a classic, Topoi has fortunately been reprinted by Dover as a cheap paperback.