Questions vary so widely there really isn't a single method that always works. I will say that there are two distinct fields of 'questions', each of which can be attacked in a manner I will attempt to -very- broadly outline.
Case 1. This is the bane of the analytic philosopher. The question that is too ill defined to answer sensibly without making a whole rash of assumptions playing more to human psychology than to empirical fact. "What is the meaning of life" is probably the one spouted about the most. I think on this sort of question one finds someone like Camus more helpful than someone like Kant (though with folks like Dennett, there could be a bit of technical insight about natural tendencies that might come to bear light).
Camus does not attempt to provide a solid answer or concept that can be utilized in understanding an external reality. Camus' writings contain psychological insight, and the same is true of Nietzsche, Schopenhauer, Sartre, Dostoevsky and many others. One aspect of philosophical investigation is the search to understand yourself and the social context in which you exist.
Case 2. This is the joy of the analytic philosopher. The class of questions that interplay with and clarify aspects of and topics in mathematics computer science, linguistics, Neuroscience ect. When a question is very broad or ill defined, the analytic philosopher can step in and decompose it into a rational question with a clear (but often hardwon) answer. I would say that 'What is a good working definition of "effectively computable", and how can we determine what is not effectively computable', when it was first asked and seriously pursued in the early 20th century, was very much a question of analytic philosophy. There was no computer science then; that came about shortly thereafter.
I would claim, generally, that analytic philosophy is ideally the first stepping stone to a legitimate field of science. The tradition is old, from Aristotle to Leibniz/Newton to Turing/Godel and on. Analytic philosophy provides a forum in which we can fumble around with weak bearings on what it is we want to accomplish, with the ultimate goal of applying our new-found understanding to the body of human knowledge in a refined form that is usable in everyday tasks.
Case 1 Section 2: The Method of Attack.
There is not much to be said here beyond that what is required is a great deal of honest self exploration and interaction with those around you. Combined with this, you should liberally apply sense and reason to what you are observing, and with experience you can draw reasonable, and sometimes very cutting and insightful conclusions. Coincidentally this is also the playground of the rank amateur, as it is very relevant to coming to a sense of self.
Case 2 Section 2: Method of Attack.
So much has been written on how to systematically attack problems this seems fairly redundant. You try to use your intuition to give a reasonable interpretation (or sometimes many) and then once you have working definitions and a fairly clear outline of the problem, you think about how you might break it down into smaller parts. You tackle each part of the problem, possibly dividing it into digestible subparts and you use your intuition and ability to analyze and attack the subparts from various angles. Some will fall easily, some may be insurmountable, depending of course on the depth of the problem and your facility with reason and your breadth of relevant knowledge.
There is no clear strategy to solving the small problems outside of becoming fluent with the process of analysis and practice with reasoning skills. It has been extraordinarily helpful for me to tackle difficult subject matter that forces me to think logically and creatively at the same time such as mathematics, logic and computer science.