Reddit mentions of Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften (322))

A couple good ones to get started are:

• A Classical Introduction to Modern Number Theory - Ireland & Rosen
  
  
• Primes of the form x^(2)+ny^(2) - David Cox
  
  These will introduce a lot of the main ideas in a way that you can understand with only the basics of abstract algebra. Having background in Galois Theory helps (you can use Dummit & Foote for a solid intro). Cox's book is probably one of the best reads in the subject. Then there are a few that introduce the basic formalism rigorously:
  
  
• First two/three chapters of Algebraic Number Theory - Neukirch
  
  
• Milne's Online Notes (PDF)
  
  
• A Course in Arithmetic - Serre (PDF)
  
  These cover the basics and can be found from many sources via amazon and google. After this, you can look at some more specialized stuff, though the two main directions are Analytic Number Theory and Langlands Program:
  
  
• Analytic Number Theory: Introduction to Analytic Number Theory - Apostal
  
  
  
• Local Fields: Local Fields - Serre
  
  
• Cyclotomic Fields: Introduction to Cyclotomic Fields - Washington
  
  
• Class Field Theory: Algebraic Number Theory - Cassels & Frolich, the final chapters of Neukirch, Artin L-Functions: A Historical Approach - Snyder (PDF)
  
  
• Elliptic Curves: Rational Points on Elliptic Curves - Silverman (undergrad level), The Arithmetic of Elliptic Curves - Silverman
  
  
• Modular Forms: A First Course in Modular Forms - Diamond & Shurman, Introduction to Elliptic Curves and Modular Forms - Koblitz, Modular Functions and Modular Forms - Milne (PDF), Automorphic Forms and Representations - Bump
  
  
• Langlands Program: (The theory that ties absolutely everything together) An Introduction to Langlands Program - Various, An Elementary Introduction to the Langlands Program - Gelbart (PDF)
  
  
• Prerequisites for the Langlands Program - Knapp (PDF) - This has a basic overview of most everything encountered in algebraic number theory along with many more references. Check it out!
  
  This may be looking far into the future, for someone just getting into Algebraic Number Theory, but I figure it's good to have a reference to point people to. I have my own biases and preferences, if anyone else wants to add something, go for it!