Hume's stated method is scientific, of careful observation and inference from particular instances to general principles. The drive of scientific inquiry is to dig deeper and deeper so as to uncover a very few, very simple principles that govern all the complexities that we observe. Newton's genius gives us three very simple laws that can explain and predict all physical phenomena. Hume wishes to perform a similar feat for human understanding (the word "understanding" is used by Hume to describe most broadly the several faculties of human reason). The hope is that Hume will derive a similarly small and simple number of principles that can explain and predict the processes of human thought. His method will be to proceed from simple observation of how the mind works and how we use it in everyday life, and to infer from his observations increasingly general principles that govern our understanding until he reaches a bedrock of simplicity and clarity.
In this respect, Hume follows very much in the empiricist vein of philosophy and owes a large debt to ##John Locke##. Locke moved against rationalist philosophy, best exemplified by ##Descartes##, which relies heavily upon rational intuition. The empiricist tradition asserts that experience, and not reason, should serve as the basis of philosophical reasoning.
The motivation for Hume's project is made apparent in his complaint that the "accurate and abstract" metaphysics that he is pursuing is frequently looked down upon and disdained. The difficulty and counter-intuitive nature of these inquiries often lead to errors that may seem absurd and prejudicial to future generations. Even today, there is a great deal of debate as to whether there has been any real "progress" in philosophy: we may have refined our discussions and dismissed some bad ideas, but in essence we are still mulling over the same problems that concerned Plato and Aristotle. It would not be unreasonable to suggest that we are no nearer a satisfactory and final answer than the ancient Greeks. Hume hopes that scientific observation can uncover the principles that underlie our reasoning so that we can be more immediately aware of faulty logic and more easily guided along the correct path.
Ever since the scientific revolution of Newton, Galileo, and others, science has been held up as a paradigm of fruitful reasoning. In science, there is a carefully defined methodology that precisely details how we can test a theory and determine whether it is right or wrong. Though it is often difficult to determine the right answer, the scientific method usually prevents us from arriving at answers that are far from the mark. Philosophy lacks any such determinate method, and philosophers are continually taking up conflicting views. For instance, Hume's emphasis on observation goes directly against Descartes' rationalism, which disparages observation in favor of pure reason. Hume hopes that his empiricism will open the way for a carefully defined method that will not allow for such disparity amongst philosophers.
Hume also suggests that his work must be epistemically (epistemic: of, relating to, or involving knowledge; cognitive) prior to the new science that he so lauds. The scientific method is a product of careful reasoning, and is thus subject to the laws of human understanding. While science seems to be in far better shape than philosophy, it too can benefit from his work. In this way, Hume differs from his predecessor, Locke. Locke sees himself as laboring on behalf of the new science, clearing away some of the linguistic rubble that might lead to confusion. While Locke humbly sees himself as simply clearing a path for science, Hume believes that his own work must lay the groundwork upon which science can rest. If he can uncover the precise laws that govern our reasoning and inferences, this should help us draw the right conclusions in our scientific investigations.