I have been encountering many results on logic-related properties of algebraic structures such as elementary equivalence, axiomatizability, definability, etc. The problem is that when I see the proof or a sketch of the proof, I understand every third word or less. What should I read to get proper understaning of the methods used in such proofs? I would most apperciate a book focused mainly on algebra and first order logic.
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A useful (and free!) book is Fundamentals of Model Theory by Weiss and D'Mello.
I second the recommendation for Hodges' Model Theory. It is long-run more useful than his Shorter Model Theory.
Marker's Model Theory is also very good.
An excellent choice is Wilfred Hodges textbook Model Theory, which goes into much further detail than most alternatives, and has many illuminating notes that will help you to "think like a model theorist" (and some good jokes to boot!) Also extremely useful are the many survey articles in the Handbook of Mathematical Logic (edited by J. Barwise)