By ordinal logic, I mean turing's ordinal logic.
I'm going to learn first order logic, elementary set theory, basic computability, and godelian incompleteness as prerequisites for ordinal logic. But, since I'm a math beginner, I don't know if I will need model theory to understand it.
Perhaps not a lot of model theory, but some. The basic results known to Turing when he did that work probably come into play, and in any case will serve you well as general culture: the completeness theorem and the compactness theorem, and possibly the Lowenheim-Skolem theorem.
The completeness theorem in its most basic form says that a sentence of first order logic is provable iff it is true in all models. Provability is, of course, relative to some particular deductive system: Hilbert-style systems consisting of axioms and rules of deduction, Gentzen's sequent calculus, the later "natural deduction" systems — these are all equivalent.
A more general form of the completeness theorem states: a theory $T$ (a set of first-order sentences) is consistent iff it has a model. Consistency of $T$ means that no contradictions can be proved using the sentences of $T$ as axioms.
The compactness theorem says that a set of first order sentences S is consistent iff every finite subset of S is consistent.
For languages with countably many non-logical symbols, the (downward) Lowenheim-Skolem theorem states that if a theory $T$ in such a language has a model, then it has a countable model. There is also an upward Lowenheim-Skolem theorem, which says that if a theory has an infinite model, then it has models of all infinite cardinalities greater than that of the model. The theorem reveals a limitation of first-order logic: a first-order theory cannot characterize its models uniquely, up to isomorphism.