Other answers to this question have been written by Chris Ferrie (more introductory than this page) and Xiaodi Wu (more comprehensive than this page).

General background: Quantum computing (theory) is at the intersection of math, physics and computer science. (Experiment also can involve electrical engineering.) Eventually you will want to learn aspects of all of these fields, but when starting you can use any for an entry into the field. Within each field, the subjects you will want to know are:

General quantum computing texts: Here is a very partial list of resources for learning more about quantum computing and quantum information.

The canonical reference for learning quantum computing is the textbook Quantum computation and quantum information by Nielsen and Chuang. Another good book (with more of a "little yellow book" experience) is Classical and Quantum Computation by Kitaev, Shen and Vyalyi.

Online resources: There are also great free resources online.

If you want to get a flavor of what research is currently hot, then one place to look is at the program of the last few QIP workshops. A less curated list of interesting papers can be found at scirate.com, where looking at the most scited papers in the last year should bring up some interesting work.

Specialized sources: Some more specialized books/lecture notes are here. These are more modern and in-depth than the general resources above.

If you have more resources to suggest or any comments on this page, then please email me at aram@mit.edu.