This is the easiest book on the subject that I found. It should provide a very excellent foundation for acquiring a very good understanding of all of the underlying ideas. It has a whole chapter on type theory and it weaves set theory in here and there as needed.
Formal Semantics: An Introduction (Cambridge Textbooks in Linguistics) by Ronnie Cann
This book provides a clear and accessible introduction to formal, and especially Montague, semantics within a linguistic framework. It presupposes no previous background in logic, but takes the student step-by-step from simple predicate/argument structures and their interpretation through to Montague's intentional logic. It covers all the major aspects, including set theory, propositional logic, type theory, lambda abstraction, traditional and generalised quantifiers, inference, tense and aspect, possible worlds semantics, and intensionality. Throughout the emphasis is on the use of logical tools for linguistic semantics, rather than on purely logical topics, and the introductory chapter situates formal semantics within the general framework of linguistic semantics. It assumes some basic knowledge of linguistics, but aims to be as non-technical as possible within a technical subject. Formal Semantics will be welcomed by students of linguistics, artificial intelligence and cognitive science alike.
I can personally attest to the excellence of this book it explains all of the key concepts in a way that is much easier to understand than most other sources, such as Monatague's original papers.