Reddit mentions of Algebraic Number Theory

Intoduction to Langlands Program you can find downloadable pdfs online. This is a fair overview of everything, not many rigorous proofs, but with something as ambiguous as Langlands the general idea is what is important.

To be able to tackle it, you're going to need to know pretty much all math up to second year graduate courses, especially Measure Theory, Representation Theory, Group Cohomology and Harmonic Analysis. You will then need, in addition to that, Class Field Theory and Tate's Thesis. These can both be found in Cassells and Frolich, though a more readable Tate's Thesis is also in the first book. This is just the standard reference book for Class Field Theory, I would recommend looking at Primes of the Form x^2 +ny^2 first. It is written at the advanced undergrad/beginning graduate level, but it is a very excellent read (probably my favorite book) and gives clear motivation for Class Field Theory which is where other books usually fail. It's good to have other sources too, like Neukirch or Lang.

TL;DR You need all of math to talk about Langlands.