TLDR: I'm interested in (algebraic) number theoretic topics, such as complex multiplication, and class field theory. As a beginning undergraduate student, how do I make my way through number theory?
My interest in number theory began indirectly with my discovery of elliptic curves, but having learned about their relationships with strange things called modular forms, my interest only grew. About a year later, I've peeked into and established interest in algebraic number theory, and class field theory. In reality, however, I hardly understand what these fields actually refer to; I figure that here would be the best place to get an idea of what these fields actually deal with, so here are my questions:
-Through analogy, one can roughly state that "multivariable calculus and linear algebra lead into classical differential geometry". Another, abstract algebra can lead into Galois theory. Where does "elementary number theory" (basic problems and theorems about prime numbers, congruences, etc...) lead?
-More precisely, a pure mathematician might highlight modern number theory in terms of Diophantine geometry, algebraic number theory, and analytic number theory. Is this fair (for someone who isn't particularly interested in analysis and raw calculations of primes), and what actually is algebraic number theory? What is class field theory? What are the primary objects (algebraic structures) studied in the major fields of number theory? What are their major results?
-Finally, how should one begin learning number theory in order to build up to a book such as David Cox's Primes of the Form $x^2+ny^2$?
I've taken a liking to Ireland and Rosen's text, but it requires a bit stronger of a command of algebra than I have. I've also found myself fond of Hardy and Wright's book. Is there a source you know of that teaches number theory and algebra at the same time? Perhaps two texts that work very well together?
Thank you so much.