Read all about it!

There are tons of books on algebraic number theory out there (some of which don't assume very much from classical number theory). Some titles that come to mind (with commentary):

• Esmonde and Murty, Problems in Algebraic Number Theory (beware of the many small errors, hopefully to be corrected in a future edition)
• Ireland and Rosen, A Classical Introduction to Modern Number Theory (a pretty good read, I'm told)
• Lang, Algebraic Number Theory (not an easy read, assumes undergraduate algebra)
• Marcus, Number Fields (mostly does examples)
• Neukirch, Algebraic Number Theory (not to be confused with his other books, which are very difficult reading)
• Pollard and Diamond, The Theory of Algebraic Numbers (the avoidance of abstract algebra makes this an easy read but obscures certain points)