What would be a study path for someone in the level of Hartshorne's Algebraic Geometry to understand and study interuniversal Teichmuller (IUT) theory? I know that it heavily relies on anabelian geometry and earlier works of Mochizuki, but what's the order to study those material? I think I had seen somewhere a complete list of papers to read from beginning to end in order to come to a level of understanding to tackle the original four papers about IUT theory, but I can't find it.

5$\begingroup$ One of Mochizuki's progress reports has a list of what parts of which earlier papers you need to read to understand the IUT series. $\endgroup$ – S. Carnahan♦ Sep 2 '15 at 0:21

2$\begingroup$ So you're saying that what components from this list and more? kurims.kyotou.ac.jp/~motizuki/papersenglish.html $\endgroup$ – Suvrit Sep 2 '15 at 1:29

$\begingroup$ @Suvrit Yes from those or from anywhere else. Simply, what I need to read before tackling IUT Theory. $\endgroup$ – terett Sep 2 '15 at 10:54
According to Mochizuki himself, the essential prerequisites for the IUTeich papers are:
Semigraphs of Anabelioids (sections 1 to 6)
The Geometry of Frobenioids I: The General Theory (complete)
The Geometry of Frobenioids II: PolyFrobenioids (sections 1 to 3)
The Etale Theta Function and its Frobenioidtheoretic Manifestations (complete)
Topics in Absolute Anabelian Geometry I: Generalities (sections 1 and 4)
Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms (section 3)
Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (sections 1 to 5)
While other sources also recommend:
The HodgeArakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories
The GaloisTheoretic KodairaSpencer Morphism of an Elliptic Curve
Particularly interesting is Fesenko's recent extended remarks on IUT (and learning IUT):
 Ivan Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions
There's also an introductory paper by Yuichiro Hoshi, but at least for the moment it is avaible in japanese only
 Yuichiro Hoshi, Introduction to interuniversal Teichmüller theory
As for the (considerable) gap between Hartshorne and Mochizuki's work, the references on each paper are quite concrete and helpful (see for example the ones on Topics in Absolute Anabelian Geometry I for a good sample).
I'll try to cover the "considerable gap" mentioned in Myshkin's answer, from algebraic geometry on the level of Hartshorne to Mochizuki's work. As a disclaimer, I'll mention that my understanding of Mochizuki's approach is very, very far from satisfactory; however, I believe that it's still possible to figure out some of the more basic prerequisites to Mochizuki's work (which at the moment I'm also still trying to learn about) without entirely understanding Mochizuki's work itself.
"0". Algebraic Number Theory  Aside from algebraic geometry, some basic ideas of modern algebraic number theory are certainly important, such as the localglobal principle, and why we might want to develop analogues for padic fields of things we are already familiar with in the real and complex case.
Suggested references: Algebraic Number Theory by Jurgen Neukirch or Algebraic Number Theory by J. W. S. Cassels and A. Frohlich.
I. Elliptic Curves  Elliptic curves play a big part in Mochizuki's approach. The abc conjecture is equivalent, via the Frey curve, to the Szpiro conjecture, which concerns elliptic curves. Mochizuki's approach is often said to be (including by Mochizuki himself) analogous to proofs of the Szpiro conjecture for the function field case; first, in his earlier "HodgeArakelov Theory", he tries to follow the proof of Szpiro himself (see Minhyong Kim's answer to this question), and in his "InterUniversal Teichmuller Theory", the proof of Bogomolov and Zhang (see Zhang's paper, and Mochizuki's discussion of the analogy).
In addition, the Tate parametrization of elliptic curves plays an important role in Mochizuki's InterUniversal Teichmuller Theory. See, for example, Kiran Kedlaya's lecture at the 2015 conference on InterUniversal Teichmuller Theory at Oxford.
Suggested References: The Arithmetic of Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves by Joseph Silverman
II. Anabelian Geometry  Anabelian geometry is the idea that we can "reconstruct" certain algebraic varieties (called anabelian varieties) from their etale fundamental groups. Before developing InterUniversal Teichmuller Theory, Mochizuki became wellknown for proving that hyperbolic curves (which include, for example, elliptic curves with one point removed, and the projective line with three points removed) are anabelian varieties. Ideas of "reconstruction" figure heavily into Mochizuki's approach; although in what way exactly, I have not yet been able to figure out.
Suggested References: The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves by Hiroaki Nakamura, Akio Tamagawa, and Shinichi Mochizuki
Additionally, I second the recommendation of Fesenko's notes mentioned in Myshkin's answer, as well as notes from this seminar on Kummer Classes and Anabelian Geometry at Vermont.