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Montague semantics assigns denotations to natural language expressions compositionally using lambda calculus and higher-order logic. Every syntactic phrase receives a semantic translation, and sentence meaning is a function of parts and grammatical arrangement. It demonstrates that natural language has mathematically precise semantics despite apparent irregularities, unifying syntax and semantics.
You already know that lambda calculus lets you define anonymous functions and reduce them through beta-reduction, and that first-order semantics assigns truth conditions to sentences via models containing a domain of individuals. Montague semantics brings both tools together to handle the full complexity of natural language — including quantifiers, relative clauses, and intensional verbs that resist simple first-order treatment.
The central commitment is compositionality: the meaning of any expression is determined by the meanings of its immediate parts and the syntactic rule used to combine them. This sounds obvious, but it has a powerful consequence — if you assign every lexical item a semantic type and every syntactic rule a corresponding semantic operation, the meaning of infinitely many sentences follows automatically. Montague showed this could be done with full mathematical rigor in his fragment of English.
Types are the machinery that makes composition work. Individuals have type `e`; truth values have type `t`; intransitive verbs denote functions from individuals to truth values, type `<e,t>`; transitive verbs are `<e,<e,t>>`; and so on. Quantificational noun phrases like "every linguist" require a still higher type: they denote generalized quantifiers, functions from sets (properties) to truth values, type `<<e,t>,t>`. This type-lifting move is one of Montague's most important contributions — it gives NPs a uniform semantic treatment regardless of whether they are names or quantifiers.
Where lambda calculus enters is in building complex denotations from simpler ones. A transitive verb like "loves" denotes a curried function; partial application to an object argument (like "Mary") produces a VP denotation; further application to the subject yields a sentence. Each syntactic rule has a corresponding semantic rule, and the derivation tree drives both parsing and interpretation simultaneously. When you write out a Montague-style derivation, you are doing syntax and semantics in lockstep — the grammar is simultaneously a phrase-structure grammar and a system of semantic equations.
One important limitation to understand: Montague's original system handles extensional semantics cleanly but requires intensional logic (possible-worlds semantics) to handle attitude verbs like "believes" or modal auxiliaries like "might." This is why the system is built on higher-order intensional logic rather than plain first-order logic. The prerequisites you bring — lambda calculus and first-order semantics — give you the two pillars; Montague semantics is what happens when those pillars are assembled into a coherent formal architecture for natural language.