There is an excellent book I read on this subject called Why Programs Fail, which outlines various strategies for finding bugs ranging from applying the scientific method to isolate and resolve a bug, to delta debugging. The other interesting part of this book is that it does away with term 'bug'. Zeller's approach is:
(1) A programmer creates a defect in the code.
(2) The defect causes an infection
(3) The infection propagates
(4) The infection causes a failure.
If you want to improve your debugging skills, I highly recommend this book.
In my own personal experience, I've found plenty of bugs in our application, but management simply presses us onwards to get new features out. I've frequently heard "We found this bug ourselves and the client hasn't noticed it yet, so just leave it until they do". I think being reactive opposed to proactive in fixing bugs is a very bad idea as when the time comes to actually put a fix in, you've got other issues that need resolved and more features management want out the door ASAP, so you get caught in a vicious cycle that can lead to a great deal of stress and burn out and ultimately, a defect ridden system.
Communication is also another factor when bugs are found. Sending an email out or documenting it on the bug tracker is all fine and well, but in my own experience, other developers find a similar bug and rather than reuse the solution you put to fix the code (as they've forgotten all about it), they add their own versions, so you've got 5 different solutions in your code and it looks more bloated and confusing as a result. So, when you do fix a bug, make sure a few people review the fix and give you feedback in case they have fixed something similar and found a good strategy to dealing with it.
limist mentioned the book, The Pragmatic Programmer which has some interesting material on fixing bugs. Using the example I gave in the previous paragraph, I'd look at this: Software Entrophy, where the analogy of a broken widow is used. If two many broken windows appear, your team may become apathetic towards ever fixing it unless you take a proactive stance.