Lots of advice: http://www.mathcs.emory.edu/~dzb/advice.html
The first 2 1/2 chapters of Neukirch to contain most of the core material. Neukirch has great exposition, but not too many problems, so I would supplement this with problems from Marcus’s Number Fields book (this one has many more problems, especially computational ones). Keith Conrad has quite a few good notes. See also this MO question.
Its also important to know the statements of class field theory and how to apply them in various situations. (The proofs, while beautiful, aren’t what I’d consider core background.) On the other hand, you do need to know Galois cohomology, adeles, statements of proofs, etc.