Matt Leifer once told me people should start (for a "conceptual" background) with John Polkinghorne's Quantum Theory: A Very Short Introduction. Then, to delve into the concepts in depth, I would recommend getting (when it is released in a month or two) Q-PSI: Quantum Processes, Systems, and Information by Ben Schumacher and Michael Westmoreland (Cambridge U. Press). I've been using it to teach undergrad. QM for a bit over a year. I would supplement it with J.J. Sakurai's Modern Quantum Mechanics (not Advanced Quantum Mechanics - they're completely different books) and, to truly appreciate a different viewpoint, Quantum Paradoxes: Quantum Theory for the Perplexed by Yakir Aharonov and Daniel Rohrlich.
If you get interested in the quantum information side of things (which is highly mathematical), pick up Protecting Information: From Classical Error Correction to Quantum Cryptography by Susan Loepp and Bill Wootters and the "bible" of QI, Quantum Computation and Quantum Information (known popularly as "Mike and Ike") by Michael Nielsen and Isaac Chuang.