Learning roadmap for algebraic number theory
Two good books for an introduction to global algebraic number theory (i.e., number fields) are:
Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics) May 19, 2008 by Pierre Samuel (less than $8 in paperback)
A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland & Michael Rosen, Springer
Number theory is broad, and I don't think it's something one learns sequentially. (Can I say there's no royal road?) There are lots fundamental topics, such as:
- number fields
- p-adic numbers
- quadratic forms
- elliptic curves
- modular forms
- class field theory
- zeta functions
- L-functions
Most of these topics (all but CFT) you could start learning first, and there are lots of ways to learn them.
Here are some some suggestions for relatively gentle introductions to these topics, though tastes vary and you may prefer other books:
Stewart and Tall's Algebraic Number Theory (mentioned in answers to linked questions): treats number fields, with a little bit on elliptic curves and modular forms. gentle, but doesn't cover some important topics.
Serre's Course in Arithmetic: p-adics, quadratic forms, modular forms. beautiful. every number theorist should read.
Silverman and Tate's Rational Points on Elliptic curves: nice elementary treatment of elliptic curves
Kato-Kurokawa-Saito's Number Theory I, II: brief treatments of various topics such as $p$-adics, number fields, zeta functions, elliptic curves. Sequel describes class field theory without proofs.
I'm not saying you should read all of these right now, or in this order, or that reading these will give you deep enough knowledge to do serious research on these topics. I am suggesting you learn the basics of some of the major topics in (algebraic-ish) number theory. (The above list of books still don't cover all the "basics"--e.g., I'm worried Gauss' theory of binary quadratic forms will be missing and Dirichlet's theorem on arithmetic progressions maybe missing, though if you get through the above you can find this stuff in my number theory ii notes, for instance, which also take the approach of a smattering of topics). Then you can try to go deeper in what you're most interested in, or maybe you'll be in grad school with more guidance at that point.