Once you've tossed your junk mail, take all bills and place them in your bill "in box" so they're in one spot. You don't need to open the bills right away, although if it helps to manage your budget as well as your paper, open them, and recycle the extra papers that inevitably accompany your electric bill, 401K statement, or credit card bill. Keep the actual bill and the return envelopes together in your in-box until ready to pay. If you pay bills online, even better.
If you don't have a bill-paying system, start simple: Label a file with a general title, such as, "2012 House Bills." When you are ready to pay them, pull up a recycling bin or recyclable paper bag, toss the envelopes and extras out (if you haven't already), and place the paid bill stubs in your folder for yearly bookkeeping. Keep all filing simple: Your bills for the year should go in one file, unless you intend to shred them.