The usual reference here is Serre's Algebraic Groups and Class Fields. But it still felt to me like there was a lot of magic left over after I read this.
We had an earlier question along similar lines with many good answers. I remember really liking David Ben Zvi's lecture, but wishing that some one would write up the "decategorification" to get from the categorical statements in Ben Zvi's talk to concrete reciprocity laws.
Even more old answers here.