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I would like to know, what are the simplest/starting/basic examples that are typically used to introduce students to how AdS/CFT really works? (not the MAGOO paper, as I am not sure it has concrete examples that a beginning student can work through)

For example, are the PhD. papers of Maldacena used for this purpose? What about the papers on holographic entanglement entropy (like the ones by Faulkner and Hartman and Shinsei Ryyu and Takayanagi)? Or the papers on worldsheet derivation of the duality like http://arxiv.org/abs/hep-th/0703141 (and its predecessors) and http://arxiv.org/abs/hep-th/0205297 and the Gopakumar-Vafa papers of '98?

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Assuming that you are familiar to some extent with quantum field theory and supergravity, you will want to start with one of the TASI lectures on AdS/CFT, e.g. arxiv.org/abs/hep-th/0201253 –  sujeet Apr 20 '13 at 1:50
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There's also Maldacena's own notes for a TASI lecture though maybe not as "easy" arxiv.org/abs/hep-th/0309246 –  sujeet Apr 20 '13 at 1:53
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Btw, for those who are not familiar MAGOO = arxiv.org/abs/hep-th/9905111 –  sujeet Apr 20 '13 at 1:56
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The review by Aharony et al (MAGOO) isn't incredibly difficult, at least the first chapters where they discuss the case of Type IIB and N = 4 SYM. If you're having difficulties you should probably read about Dp-branes in any of the string textbooks. There's also a more explicit but less deep review, hep-th/9902131. –  Vibert Apr 20 '13 at 8:19
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@Vibert Can you say list out some of the important D-brane papers which actually derive any of the necessary ideas? Can you may be list some papers which go through a pedagogic calculation of the duality? –  user6818 Apr 21 '13 at 21:13

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One review I found helpful relates Type IIB superstrings on a maximally supersymmetric plane wave background to N=4 SYM:

Lectures on the Plane–Wave String/Gauge Theory Duality by Jan Christoph Plefka

It might be more conversational than what you're looking for, but this review by Polchinski has some great discussion and probably deserves the mention:

Introduction to Gauge/Gravity Duality by Joseph Polchinski

One very good, specific set of lecture notes already mentioned but deserving repeat is:

Supersymmetric Gauge Theories and the AdS/CFT Correspondence by Eric D'Hoker, Daniel Z. Freedman

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As I was asking Vibert above - in any of these reviews can you point to a specific pair (one in gravity one in gauge) of calculation which one can work through to convince oneself of the duality? I am looking for specific examples. –  user6818 Apr 22 '13 at 20:25

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