Learning about Grothendieck's Galois Theory.

Solution 1:

Possibly, H. W. Lenstra's Galois theory for schemes might be of interest to you

http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf

Solution 2:

In my very humble opinion, this is a very hard topic to find a solid book on. Lenstra's text is very feel-good, but has serious drawbacks. It does everything very old fashioned. For example, Lenstra thinks of curve theory in terms of valuation theory. This makes somewhat simple curve arguments, like calculating $\pi_1(\mathbb{A}_k^1)$ (characteristic zero) seem overly complicated. He also does everything in terms of pure algebra--not in the modern language at all. This makes it hard to translate results into the language you may accustomed to, and for which others speak in terms of.

Szamuely is a top-notch book, in almost all regards. It really is one of my favorite math books. It will teach you quite a bit about the basics of anabelian geometry, including discussions of more recent results. In terms of culturing and big picture, you can do no better. My only complaint is that he also doesn't really assume you know scheme theory. It's not as bad as Lenstra's pure leg bra approach, but I found it somewhat hindering at some points.

If you're looking for all the basics (definitions, background, first results) I'd suggest you look at EGA IV. Or alternatively, Lei Fu's Etale cohomology book, which essentially is a dryer and more detailed version of the relevant EGAs--in English I might add.