Jech's Set Theory logic prerequisites

At the very least, you should understand the statements of the completeness theorem, the incompleteness theorems, and the Lowenheim-Skolem theorem. This involves all of the basic definitions of formal logic and model theory; models, theories, satisfaction, elementarity, elementary substructures, Skolem functions, and more. Of course, as it is in any area of math, it's easy to trick yourself into thinking that you understand these things, when you really don't. You can get a more thorough understanding by reading the proofs, which make up the essentials of many logic texts.

I don't know the Chiswell-Hodges book, but glancing at the table of contents, it looks like it doesn't quite cover all of the prerequisites. (It may however be a very good book for what it does cover.) Enderton's book (on logic) is considered a classic, and it has approximately the right content, including the relevant definitions from model theory.

General model theory is a good thing to have an understanding of when going into set theory, but don't think it'll make the "Models of Set Theory" chapter of Jech a breeze; models of set theory are pretty counter-intuitive at first glance, and require a lot of thinking about on their own.

I agree with Arthur Fischer in the question you linked to; Jech is not a great book for a budding set theorist to learn the field from. It's more of a reference than a text. However, I think the above remarks apply to essentially any introduction to modern set theory.