A book about model theory I am looking for a good book about model theory. As this is obviously too vague, let me
explain what I am looking for and why.
First I am interested about the basics and  foundations of model theory. Right now I am not interested in their applications (like proving things in mathematics or even independence results like Cohen's -- but of course it is not a problem if a book deals with some of these applications if it does not only that). 
Second,  until a few days ago I believed I knew well enough what was a model. 
But since two days I am  not so sure. I have a problem with the notion of model inside ZF (or ZFC, or any formalized set theory) of a theory, and in particular, with the meaning of the satisfaction relation in this case. I would like a book which treats that aspect.
These problems arose with my trying to understand the answers to my question
A meta-mathematical question related to Hilbert tenth problem 
I am currently having endless discussions in comments about that notion what I am said doesn't make sense to me (and the converse is clearly also true). A good book will certainly help me and save time for my respectable interlocutors. Thanks...
 A: Basic model theory texts are Marker's Model Theory; An Introduction and A Shorter model theory by Hodges. Maybe the one on Mathematical Logic by Cori and Lascar too. I'm not sure you need a book which specifically treats this aspect but a general understanding of what a theory, and a model of a theory (e.g. ZF or ZFC) is should do (the first chapter of Marker's book covers this). Once you know the basics then just think of a language with a binary relation symbol (for set membership) and formulas (in the language) for the axioms of ZF. Then you can think of a model of this etc...
A: For understanding the referenced MO question, are you sure that you want a book on Model theory? I think that a request for a book introducing  first order logic is what this post is really asking for. I won't make suggestions for learning basic first order logic, but perhaps others could. 
In case I am wrong, and the poster is really interested in learning model theory. I don't think I see the point of answers which simply list model theory texts and have little to do with the specific reference request. Any model theory book will give the poster the basics. From the question, a book emphasizing models of PA would be a better suggestion. Here are some details about the specific books:
Marker's book covers PA in several chapters based on the model theoretic techniques used and contains a fair number of exercises on models of PA. 
Poizat's book contains much less (almost nothing technical), so it would not seem to be a good suggestion. 
From what I recall Marcja and Toffalori contains even less (although I haven't looked at this much, so I could be wrong in this recollection). 
Hodges contains less than Marker and more than Poizat, but is also not a good choice for this poster's goals. 
I have not read Cori and Lascar, so I can not comment definitely, but I will mention that the table of contents at least mentions the Peano axioms. Also, this suggestion would seem to be in line with what I wrote in the first paragraph.
The above books were included (in part) because other people mentioned them in their answers. The following should be mentioned, but this is not a comprehensive list. Hodges Model theory (not the shorter one...) has more PA than his shorter book, but still not as much as Marker's book. Kaye's Models of Peano Arithmetic is hard to find (at least it was a few years ago), but the parts of it that I have read seemed well-written. 
A: 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.
A: Brunot Poizat's "A course in model theory" or Marcja&Toffalori's "A guide to classical and modern model theory" come to mind.
If you need very very basic things about models, Cori&Lascar "Logique mathématique" has an english translation.
A: Rather than to a book, I point you to real formalizations in a set theory: I deem this appropriate given the question itself. Otherwise, downvote me please :)
I happen to have formalized in Mizar set theory (which is Tarski-Grothendieck, i.e. ZFC on steroids) the stuff you seem pointing to: language, wffs, interpretation, satisfaction relation, evaluation, sequent derivability, provability, etc...
In FOMODEL1:
http://mizar.auburn.edu/version/current/html/fomodel1.html
you get most syntax (up to definition of atomic formula, or 0wff).
In FOMODEL2:
http://mizar.auburn.edu/version/current/html/fomodel2.html
you will find the definition of satisfaction.
That is a series of five subsequent articles starting from scratch and getting to completeness theorem (and Lowenheim-Skolem, the latter only on my homepage, not submitted to Mizar people yet).
The links point to hypertextual, proof-pruned versions. For full formalizations, look for the same files with the extension .miz in that same server. 
Mizar formalizations are arguably among the most readable for the average mathematician (that's the factor that got myself started with it), that's why I thought you could find this stuff of some interest.
A: I might be completely mistaken, but I dare say you should take a look at set theory first, maye that could clarify your questions.
I'd recommend the first four chapters of Kenneth Kunen's book, at least they helped my understanding, but maybe that's not the thing you had in mind.
In particular:
Chapter 1, §14: "Formalizing the metatheory"
Chapter 4, §9: "Model theory in the metatheory"
Chapter 4, §10: "Model theory in the formal theory"
