If you want to go further in understanding this point of view, I would advise you to begin learning class field theory. It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.
The texts:
Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof
of quadratic reciprocity, and other assorted attractions.
Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.
Serre's Local Fields for learning the Galois theory of local fields
Cassels and Frolich for learning global class field theory
The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT. But it is at a higher level again.
I don't think that you will find a single text on this topic at a basic level (if basic
means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.
Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris). Certainly there are grad courses offered on this topic there on a regular basis.
Your requirements are quite stringent! As you know well, ANT is a couple of layers removed from "practice". In general, I find that the methods deriving from the development of algebraic number theory eventually lead to incomparably more applications than any of the standard ANT theorems themselves. Just a few examples that quickly spring to mind: Gauss reduction of quadratic forms → shortest lattice vectors → LLL; Dirichlet units → Minkowski geometry of numbers → convex geometry; class groups and unit groups → finitely generated abelian groups → (pick your favorite application of group theory, e.g. in abelian harmonic analysis). I would try to project this deeper idea over the immediate payoff. Also, unsurprisingly, "elementary" number theory presents more immediate applications, e.g. to cryptography and specifically, to primality testing and factorization.
But enough of philosophy! Here a few concrete applications:
Construction of codes and dense lattice packings using multiplicative groups of global fields by Rosenbloom and Tsfasman (Invent. Math. paper or see Tsfasman's survey "Global fields, codes and sphere packings").
Margulis arithmeticity theorem: not only are algebraic integers useful in constructing discrete groups, but after imposing certain conditions (an irreducible lattice in a higher rank semisimple Lie group), all of them arise in this way! These lattices have been used in constructing Ramanujan graphs and superconcentrators and to uniform distribution of points on spheres (admittedly, arithmetic of the field is rather secondary: you can get good constructions starting from a group over Z).
Lind's theorem on realizability of any Perron–Frobenius integer as the leading eigenvalues of a positive integer matrix. Since log λ is the entropy, this does have semi-practical consequences (cf the textbook of Lind and Marcus).
Lucas–Lehmer primality test, Lucas and Fibonacci pseudoprimes, Grantham's Frobenius test. This strides the border between elementary and algebraic number theory, which may actually be an advantage in an undergraduate class!
I'd be curious to know how do they jibe with your goals.
Best Answer
There are probably many such books, for instance "Fundamentals of Number Theory" by LeVeque, "Elementary Number Theory" by Bolker and "A Classical Introduction to Modern Number Theory" by Ireland and Rosen.