I am a 3rd year undergraduate math student and would like to study model theory. . I have some background with set theory, ordinals etc and also with mathematical logic.
This is purely for self study and textbooks that are clear and perhaps with a good amount of examples would be appreciated.
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A canonical choice would be Model Theory : An Introduction by Marker.
Also, if you read french, there is a more advance but marvelous book : Cours de théories des modèles (Une introduction à la Logique Mathématique contemporaine) by Poizat.
Why not take a look on the sections on model theory in the reading Guide on math logic which you can download from http://www.logicmatters.net/tyl ?
Yes, Marker's text is standard; but it is tough. The Guide suggests a few texts that bridge the gap between the entry-level model theory in standard first math logic courses and the advanced text by Marker. The headline news is that Hodges's Shorter Model Theory is a particularly good intermediate choice: but there is a lot more said in the Guide.