Basics on anabelian geometry and Grothendieck's section conjecture

if you just read the Hartshorne, your background is not enough for anabelian geometry. You need

A) To learn about etale morphismes: Hartshorne just mentions that notion in exercises.

B) to learn about the étale fundamental group Grothendieck,

before C) learning anabelian geometry proper.

Let me begin by B), because for this I think there is one best reference, which is the original one: SGA I. It is certainly a non-trivial read, but not nearly as difficult as it is often imagined by beginners. After all, it is the first SGA, made for an audience who consisted only of beginners. And it is a good initiation to the grothendieck's style of thought, of which you will need plenty if you are to learn anabelian geometry.

Before learning the étale fundamental groups, you have to learn étale morphisms, as I said in A). Here you have several choices. You can stick with SGA I, whose first chapters contain everything you need on the subject. But those chapters may be a little terse for someone who just knows the Hartshorne. Alternatively, you can find a good introduction to étale morphisms in the texts about étale cohomology, such as Milne's or Freitag-Kiehl's. And of course, you can also read the relevant chapters in EGA, which will probably lead you to read other chapters first, but if you want to work in algebraic geometry, that's no wasted time.

For C), there is much to read and I am not an expert, but for an introduction you can begin with "The Grothendieck Conjecture on the Fundamental Groups of Algebraic Curves" by Hiroaki Nakamura, Akio Tamagawa, Shinichi Mochizuki.

For the section conjecture you might look at Jakob Stix's great book

Evidence for the section conjecture in the theory of arithmetic fundamental groups Habilitationsschrift, School of Mathematics and Computer Science at the Ruprecht-Karls-Universität Heidelberg, January 2011, x+190 pages, to appear as

Rational Points and Arithmetic of Fundamental Groups. Evidence for the Section Conjecture

Springer Lecture Notes in Mathematics 2054, xx+pp.249, Springer, 2013.

"The first exposition of foundational material on the arithmetic of fundamental groups with respect to the Section Conjecture of anabelian Geometry: from the history of the subject to the state of the art of the conjecture. Numerous approaches to the Section Conjecture are discussed with open questions to stimulate future research. Assuming the basics, the more advanced chapters are self contained and can be read independently."