Based on our discussions in your other MO question, I believe that what you want to see is not a book about model theory, but a tutorial about how to formalize ordinary mathematics in ZFC, with model theory being a specific case of interest. One resource you might find useful is an article by Leslie Lamport in which he takes you through an example slowly (the Riemann integral I think). You should be able to find it by Googling "formalizing mathematics lamport". Once you get the general idea, you should be able to apply it to other cases.
The only confusing thing about model theory specifically, I think, is that in model theory one works with formal languages, which don't show up in "classical" (19th century or earlier) mathematics. But if you encode symbols as sets, and strings as sequences of symbols, then there should be no problem.
