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.